Lesson 15: More About Classical Mechanics

Introduction

Lesson 15 takes the Lagrangian picture you already have and turns it into the language that actually scales: phase space, conservation from symmetry, Hamiltonians, relativistic actions, symplectic geometry, and Liouville flow. We will be using the language of differential geometry. This Lesson is in two parts, each part takes a session.

Configuration space is the manifold of possible instantaneous positions (or generalized coordinates) of the system. A point in it tells you where everything is, but not how it is moving. The real dynamical arena is larger.

Symmetries of the action (or of the Lagrangian) produce conservation laws. Cyclic coordinates give conserved momenta immediately; continuous symmetries more generally give Noether charges. Energy conservation itself appears when the Lagrangian has no explicit time dependence. These statements become sharper and more geometric once we pass to the Hamiltonian side.

The Hamiltonian is the Legendre transform of the Lagrangian with respect to the velocities. It is a function on phase space (coordinates plus conjugate momenta) and, for ordinary mechanical systems, equals the total energy. Hamilton’s equations then replace the second-order Euler–Lagrange equations with a first-order system that treats q and p symmetrically.

Before we stay in that language, we pause to write the action principle in Minkowski spacetime. The free-particle action is (minus a constant times) proper time, where the physical world-line extremizes it. This is the cleanest way to see that the principle of stationary action is already relativistic, and it sets the pattern for later field theories.

Hamiltonian mechanics is the resulting first-order dynamics on phase space. Time evolution is the flow generated by the Hamiltonian vector field. That flow has a very special geometry.

The symplectic geometry of phase space is that geometry. The canonical two-form is closed and non-degenerate; it turns functions into vector fields. Hamiltonian vector fields are exactly the ones that preserve ω.

Liouville’s theorem is the immediate global consequence: Hamiltonian flow preserves phase-space volume (in fact it preserves the symplectic form itself). Densities are incompressible. That fact is the starting point of classical statistical mechanics and of many numerical integrators.

Finally we do the algebra, the Poisson brackets where formal machinery becomes something you can actually run rather than only write on a blackboard.

The through-line is that Lagrangian mechanics is efficient for writing equations, while the Hamiltonian-plus-symplectic picture is the one that reveals structure, conservation, and geometry. That structure is what we will need when the configuration space becomes a space of fields and the constraints become the Gauss law or the diffeomorphism constraints.

Session 1

Configuration Space

The configuration space Q is the manifold whose points are the instantaneous arrangements of the system that the constraints still allow. A single point q∈Q answers only the kinematic question “where is everything?” It does not yet carry velocity or momentum. All of the dynamical structure we need sits in bundles over Q.

A fiber bundle π:E→Q is a space that looks locally like a product U×F but may be twisted globally. The base is configuration space; the fiber is the set of extra data that can be attached to the configuration q. In mechanics the two bundles that matter first are vector bundles, and then we have tensor bundles.

The tangent bundle T Q→Q has fiber , the vector space of velocities (or infinitesimal displacements) at q. A point of T Q  is a pair . That is the natural arena of Lagrangian mechanics, where the Lagrangian is a function

(15.1)

The projection T Q→Q simply forgets the velocity and remembers the configuration.

The cotangent bundle has fiber , the dual space of one forms at q. A point of is a pair (q,p). Momenta live here because the Legendre transform

(15.2)

sends a velocity in to a one form in . The same projection now forgets the momentum and retains only the configuration.

Both T Q and are examples of tensor bundles over Q. More generally one can form the bundles of tensors of type (k,l),

(15.3)

Vector fields, one form fields, and higher-rank tensor fields on configuration space are sections of these bundles. Recall that a section is a consistent choice of a point in each fiber. The bundle language keeps the distinction clean, where a configuration is a point of the base; a geometric object attached to configurations is a section of a tensor bundle over that base.

Coordinates make the same picture look elementary. A chart on Q induces linear coordinates on each tangent fiber and dual coordinates on each cotangent fiber. Changing charts on the base induces the usual Jacobian transformations on the fibers—exactly the transformation law of tensors. That is why generalized coordinates may be chosen freely on Q while objects built from and still transform in a definite way.

Holonomic constraints simply cut Q down to a submanifold; the tangent, cotangent, and tensor bundles are then those of the reduced configuration space. When the constraints act inside the fibers they restrict which vectors or one forms are admissible at each q. In either case the fiber-bundle description of configuration space says precisely which bundle is being restricted and which geometric object—Lagrangian on T Q, momentum one-form in , or a more general tensor field—is under discussion.

In short, we have Q as the list of possible shapes. , and the tensor bundles built from them are the spaces that turn those shapes into objects a variational principle can act on.

Invariance and Covariance

A geometric object on configuration space is not a list of numbers in a chart. It is a section of a tensor bundle, or a function built from such sections. Charts merely display that object. What matters is which features of the display are accidents of the labeling and which features survive every relabeling.

Some quantities are attached directly to configurations, or to kinematic states, as single numbers. A real function f:Q→R is of this kind, where if and are two charts, then

(15.4)

so the number sitting at a given configuration does not care what labels were used. The same requirement lifts to the tangent bundle. A Lagrangian L:T Q→R is a function of position and velocity. Under a change of coordinates q|→q'(q) the velocities must transform as tangent vectors,

(15.5)

where we require,

(15.6)

The numerical value of L at a given kinematic state is therefore chart-independent. Quantities with this property are called invariants.

Other objects are not single numbers. They assign to each configuration; a vector, a one form, or a higher-order tensor—that is, they are sections of T Q, , or . In a chart a vector field X has components . In another chart those numbers change,

(15.7)

A one form field changes with the inverse Jacobian,

(15.8)

A tensor field of type (k,l) picks up one Jacobian factor for each contravariant index and one inverse Jacobian for each covariant index. The components move; the section does not. An object whose components change by this law, so that they glue to a single well-defined section, is called covariant.

The two properties lock together. Contracting a covariant tensor with the right number of vectors and one-forms produces an invariant. If a putative list of components fails the transformation law, contractions yield chart-dependent junk, and there was never a section of a tensor bundle to begin with.

On overlapping charts the same story is the gluing rule for the bundle. The transition functions of T Q and are exactly the Jacobians and inverse Jacobians above. Local component functions define a global section if and only if they transform in this way on overlaps. Chart-independent values and correctly transforming components are therefore not extra physical postulates; they are what it means for the Lagrangian, the velocities, and the momenta to live on well-defined bundles over Q .

Any diffeomorphism of configuration space, or any change of generalized coordinates, must leave the invariants alone and must drag the components along by the tensor law. That is the precise sense in which the description of the system does not depend on how one labels the configurations.

Figure 15.2: The number does not care about the labels. The components do. A bad transformation law fails to glue.

Geometry of Motion in Configuration Space

A motion is not first an equation. It is a curve in the base. One specifies an interval of time and a map q:I->Q , t|→q(t), that sends each instant to a configuration. The image of this map is the path the system traces through the list of possible shapes. Everything else—velocity, momentum, the value of the Lagrangian—is data attached to that path.

Start with a curve of configurations, q(t) in Q , and a bundle π:E→Q sitting over that same base. At each instant the curve occupies some point q(t), and over that point there is a whole fiber of extra data. One may choose, instant by instant, one element of that fiber, and let the choice itself move with time. The result is a new curve that lives in the total space E, not in Q , but that always sits over the original curve, project it down and you recover q(t) exactly.

That new curve is called a lift of q(t) to E.

Differentiating the curve produces, at each instant, a vector in the tangent space of the point the curve occupies, . The pair is then a curve in the total space of the tangent bundle. That lifted curve is the kinematic history, not only where the system is, but how it is instantaneously leaving that configuration. The Lagrangian is evaluated along the lift, , and the action is the integral of that function of time. The variational principle compares nearby curves in Q by comparing the numbers obtained from their lifts in T Q .

An infinitesimal change of path that keeps the same endpoints is, at each time, a vector based at q(t). Such a vector is a virtual displacement. Because it lives in , it is a section of the tangent bundle pulled back to the curve. If the admissible configurations have already been cut down to a submanifold Q'⊂Q, every virtual displacement must lie in . Holonomic constraints are therefore not extra forces on the motion; they are a restriction of the base, and hence of every tangent space along the curve.

Figure 15.3: The path is a curve in the base. Velocity is the lift of that curve into T Q . Acceleration is deliberately not drawn.

First derivatives along a curve are geometrically clean, where is a tangent vector, and under a change of generalized coordinates it transforms by the Jacobian, as any section of T Q must. Second time derivatives are not. The coordinate functions mix, under a chart change, with first derivatives of the Jacobian. A raw list is therefore not the component list of a tensor at q(t). Comparing the velocity at t+dt with the velocity at t requires a rule for transporting a vector from one tangent space to a neighboring one. Until that rule is supplied, acceleration is not a section of a tensor bundle over Q . The geometry of motion at first order lives on T Q ; the geometry of motion at second order needs extra structure on the bundle.

None of this depends on how the points of Q are labeled. A reparametrization of the coordinates drags the curve’s component functions along by the chain rule, drags the velocity components by the Jacobian, and leaves invariant any scalar built from the lift—most importantly the value of L at each instant. The path in the base, the lift in the tangent bundle, and the virtual displacements along the path are the covariant content of a motion in configuration space.

Basis Vectors in Configuration Space

A tangent space is a vector space, so it admits bases, but nothing in the definition of Q hands you one. A chart does. Around a configuration q choose coordinates . Each coordinate is a real function on a neighborhood of q. Differentiating a curve while holding all but one coordinate fixed produces n vectors at q, one per coordinate. Those vectors span and are linearly independent. Written in the usual operator notation they are

(15.9)

This ordered list is the coordinate basis of the tangent space associated with the chart.

Figure 15.4: The chart supplies the basis. Components are coefficients in that basis; the pairing is a scalar.

Any tangent vector at q is then a unique linear combination of these basis vectors. In particular the velocity of a curve q(t) expands as

(15.10)

The numbers are not themselves vectors; they are the components of a vector relative to the basis the chart supplied. The tangent lift is exactly this expansion, recorded as a point of T Q . The dual story lives in . Each coordinate function has a differential , and these one-forms form the basis dual to the coordinate basis

(15.11)

A general one-form at q is , and the numbers are its components. The pairing of a velocity with a one-form then collapses to the ordinary sum and is a chart-independent number.

Change the chart and both bases move. If , the new coordinate vectors are linear combinations of the old ones,

(15.12)

while the new coordinate one-forms pick up the inverse Jacobian,

(15.13)

That is the same transformation law already required of components of sections. The vector and the one-form α do not change; only the bases used to read their components change. A list of numbers is meaningful as a vector or one-form only after one names the basis it is taken with respect to.

Nothing forces one to use a coordinate basis. Any smooth choice of n independent vectors at each point is a frame, that is, a local basis of sections of T Q , and it has a dual frame in . Coordinate bases are merely the frames that come for free with charts. They are the ones in which the tangent lift of a curve has components equal to the ordinary time derivatives of the coordinate functions—the reason they dominate explicit calculations.

So the geometry is this, each chart trivializes the tangent and cotangent fibers by supplying a basis and a dual basis. Vectors and one-forms are sections; their components are the coefficients in those bases. Motion in configuration space is a curve in Q ; its first-order geometry is the expansion of   in the basis the chosen coordinates provide.

Canonical Momenta and Covariance of Lagrange’s Equations in Configuration Space

The Lagrangian is a single number attached to each kinematic state, that is, a real function L:T Q→R. Under a change of generalized coordinates the velocities transform as tangent vectors and the value of L is required not to change. Differentiating that invariant function with respect to the velocity components therefore cannot produce another invariant. It produces one number per coordinate, , and those numbers inherit the transformation law of a one-form. If q|→q'(q), then

(15.14)

The list is consequently the component list of an element of . That one-form is called the canonical momentum associated with the chart, and one writes

(15.15)

The pairing of this one-form with the velocity is the invariant . The map that sends a point of the tangent bundle to the one-form p at the same q is a lift of the configuration curve into , once a motion q(t) has been chosen and its tangent lift formed.

Lagrange’s equations are a statement about that one-form along a motion. Given a curve q(t) and its tangent lift, form

(15.16)

Each of the two terms on its own is chart-dependent in a messy way. Their difference is not. Under a change of coordinates on Q the combination transforms by the same inverse Jacobian that acts on

(15.17)

The left-hand side is therefore the component list of a one-form along the curve. Setting it to zero in one chart sets it to zero in every chart. That is the covariance of Lagrange’s equations, they are not n unrelated scalar conditions that happen to look similar in every coordinate system. They are the single geometric statement that a particular section of the pulled-back cotangent bundle vanishes along the motion.

The same conclusion can be read off the variational principle. An invariant action

(15.18)

is varied by a vector field along the curve that vanishes at the endpoints. The first variation is the pairing of that vector field with the one-form whose components are the Euler–Lagrange expressions. If the pairing is zero for every such vector field, the covector itself is zero. No chart was used in the statement; charts only extract components.

Thus the momenta exist because L depends on velocities, and they are one-forms because L is a scalar on T Q . Lagrange’s equations exist because the action is stationary, and they are covariant because the failure of stationarity is itself a one-form on configuration space.

Symmetries and Conservation Laws

Cyclic Coordinates

Cyclic Coordinates and Constancy of Conjugate Momenta on the Motion

A coordinate chart on Q is a labeling, not a piece of physics. Occasionally the labeling is adapted to the variational data in a special way, where one of the coordinate functions is simply absent from L. Velocities may still depend on that label’s time derivative, and the other coordinates may appear in any fashion, but L does not change when that one coordinate is shifted while everything else is held fixed. In a formula,

(15.19)

throughout the relevant region of T Q , while need not vanish. A generalized coordinate with this property is called cyclic.

Lagrange’s equation in that slot then loses its right-hand side. Along any curve one still forms the canonical momentum

(15.20)

this is the corresponding component of the one-form in . The equation of motion reduces to

(15.21)

Thus is the same number at every instant of every motion. Constancy here is constancy on the motion, where may still depend on the remaining coordinates and velocities when it is viewed as a function on T Q , but once a solution q(t) is substituted, the composite function does not change.

The conclusion is chart-dependent in appearance and chart-independent in content. If is cyclic in one coordinate system, a generic change of labels will mix into the new coordinates and the cyclic character will disappear from the component list. The underlying fact survives that there is a direction in Q along which L is insensitive, and the pairing of the canonical momentum with that direction is an invariant of the motion. In the adapted chart that pairing is simply .

Because Lagrange’s equations transform as a one-form, it is enough to read the conservation law in the adapted chart. If there, the same geometric statement holds after any relabeling, even if no single new component of p remains constant by itself.

The practical consequence is a reduction. One first integral is known before the remaining equations are touched. The motion in Q is confined to the level sets of , and those level sets may be used to eliminate one combination of velocities from the problem that is left. A cyclic coordinate does not make a degree of freedom disappear; it makes the conjugate momentum a constant of the motion and thereby removes one quadrature from the integration.

Translation-Invariance and Conservation of the Impulse

Suppose the configurations can be described, at least locally, by Cartesian positions of the parts of the system. A simultaneous shift of every such position by the same constant vector a is then a diffeomorphism of Q , q|→q+a. Velocities are untouched, because a does not depend on time . If the Lagrangian does not notice this operation

(15.22)

for every displacement a—then no absolute origin has entered the variational data. That property is translation-invariance.

In an adapted chart the same statement is elementary. Let x be a Cartesian coordinate that is shifted by the translation. Then L does not depend on x, only possibly on and on the remaining variables. The coordinate x is cyclic. The associated component of the canonical momentum,

(15.23)

is therefore constant on every motion

(15.24)

A translation in another Cartesian direction supplies another cyclic coordinate and another constant component. The three constants are the components, in the chosen Cartesian one-form frame, of a single one-form. That one-form is the impulse.

When several bodies are present and the same vector a is added to every position, the cyclic coordinates are the components of the overall displacement and the conserved one-form is the sum of the individual canonical momenta in those directions. Translation-invariance of L is equivalent to the vanishing of the total force-like term in Lagrange’s equations, which is why the total impulse does not change.

The geometric content does not require the Cartesian chart, only the family of shifts. Each constant vector a determines a vector field on Q whose flow is the translation by s a. Invariance of L under that flow makes the pairing of the canonical momentum with constant along motions. Evaluating the pairing on a basis of translations recovers the Cartesian components of the impulse. Change those axes and the components mix by the inverse Jacobian; the impulse itself, as a one-form built from p, does not.

Thus a system whose Lagrangian cannot tell where the origin was placed has, on every motion, a constant pairing between the canonical momentum and the generators of spatial shifts. In Cartesian language that pairing is the conserved impulse.

Invariant Submanifolds and Conservation of Momentum

A conservation law does not first announce itself as a name. It announces itself as a restriction on where a motion is allowed to go. Once a particular solution q(t) is under way, every invariant built from the configuration, the velocity, and the canonical momentum takes a numerical value at the initial instant and keeps that value thereafter. The set of kinematic states at which the invariant equals that number is therefore a trap, where the tangent lift of the motion cannot leave it.

When the invariant is a component of canonical momentum the trap is especially clean. Let be the pairing of the canonical momentum with a vector field ξ that generates a symmetry—most concretely, when is cyclic, or ξ is a generator of translations when L is translation-invariant. Then

(15.25)

so the lift remains inside a single level set

(15.26)

Where is nonzero this level set is a hypersurface in the tangent bundle. A subset of T Q that contains the tangent lift of every motion that meets it is called an invariant submanifold. Each is of that kind.

Figure 155: Once at the initial instant, the tangent lift cannot leave that level set. The base Q is not cut; the fibers of admissible velocities are.

Several independent conserved pairings cut further. If translations in three Cartesian directions are symmetries, the three components of the impulse are constant, and the motion in T Q is confined to the common level set of those three functions. The same holds for any collection of cyclic coordinates, where each constant is one equation on the velocities at each q, and the simultaneous level set is an invariant submanifold of lower dimension.

The base Q itself need not be carved by these equations. A conserved momentum is typically a condition on , not on q. What is reduced is the bundle of admissible velocities. At each configuration the fiber is replaced by the affine subspace of vectors whose pairing with the relevant one-form (or one-forms) equals the prescribed constant. The invariant submanifold is the union of those restricted fibers.

Because the construction uses only the scalar L and the covariant pairing , the submanifold does not depend on how Q is labeled. In one chart it is “”; after a change of coordinates the same set is described by a different combination of components. Motions neither know nor care. They stay on because Lagrange’s equations force , and that statement is already known to be chart-independent.

The use of the picture is reduction. Instead of searching for curves in the whole of T Q , one searches inside a single invariant submanifold fixed by the observed values of the conserved momenta. Cyclic coordinates and translation-invariance are the usual ways of producing those values. Conservation of momentum is the statement that the tangent lift never leaves the corresponding .

Rotation-Invariance and Conservation of the Angular Momentum

A rotation about a fixed axis through a chosen origin moves every Cartesian position to a new Cartesian position and therefore induces a diffeomorphism of Q . An infinitesimal rotation about a unit axis n displaces each position in the direction . The corresponding vector field on configuration space is the section of T Q whose value at a many-body configuration is that collection of infinitesimal displacements. Finite rotations are the flow of .

If the Lagrangian is insensitive to the operation

(15.27)

for every rotation R, or, equivalently, if the directional derivative of L along each vanishes—then no absolute orientation of the axes has entered the variational data. That property is rotation-invariance.

In an adapted chart the same fact looks like a cyclic coordinate. Place the axis along and introduce an azimuthal angle φ  that measures rotation about that axis. Rotation-invariance means that L does not depend on φ, only on and on the remaining variables. Then φ is cyclic, and the conjugate momentum

(15.28)

is constant on every motion. The number is the component, along that axis, of a one-form manufactured from the canonical momentum and the generator . Repeating the construction for three independent axes yields three constant pairings

(15.29)

The object whose Cartesian components are these pairings is the angular momentum.

For a single body whose configuration is a position x in Cartesian coordinates, the pairing with the generator of rotations about n reduces to the familiar combination

(15.30)

where . . For several bodies the same pairing is the sum of the individual contributions. Rotation-invariance of L is equivalent to the vanishing of the total torque-like term in Lagrange’s equations associated with , that is why that pairing does not change.

As with the impulse, the conserved object is not a coordinate list. Change the axes and the three components mix; the pairings themselves, and the invariant submanifolds

(15.31)

do not. Each such level set traps the tangent lift of every motion that meets it. Full rotation-invariance supplies three independent cuts of this kind, one per axis, subject only to the algebraic relations among the generators .

Thus a system whose Lagrangian cannot tell how the axes were pointed has, on every motion, constant pairings between the canonical momentum and the generators of rotations. In an adapted polar chart one of those pairings is an ordinary cyclic momentum . In Cartesian language the three pairings are the components of the conserved angular momentum.

Emmy Noether’s Theorem

Passive and Active Transformations

There are two different operations that can produce a new list of numbers from an old list . They look similar in a formula and mean opposite things.

In the first operation nothing in the system moves. The configurations stay where they are; only the labels change. One replaces a chart on Q by another chart . Each point of configuration space acquires new coordinates, velocities acquire new components by the Jacobian,

(15.32)

and components of one-forms acquire the inverse Jacobian. The curve q(t) in the base is the same curve. The tangent lift is the same lift. The value of L at each kinematic state is the same number, as it must be if L is a scalar on T Q . What has changed is the display. An operation of this kind—relabeling points of Q without sending those points anywhere—is a passive transformation.

In the second operation the labels stay put and the system is moved. A diffeomorphism φ:Q→Q sends each configuration to a possibly different configuration. A curve q(t) is replaced by the image curve φ(q(t)), and the tangent lift is replaced by the image lift . Translations q|→q+a and rotations q|→R q are operations of this kind. They do not ask for a new atlas; they ask whether the variational data treat the displaced kinematic state the same as the original one. An operation that actually relocates points of configuration space is an active transformation.

The same Jacobian matrices appear in both calculations, that is why the two are easy to confuse. In the passive case the Jacobian compares two charts at one point. In the active case it is the derivative of a map from Q to itself, evaluated along a curve that has been moved. Covariance of components is a statement about passive transformations, where sections of tensor bundles have component lists that adjust so the section remains the same. Invariance of L under translations or rotations is a statement about active transformations, where the function on T Q takes the same value after the kinematic state has been displaced.

Figure 15.6: Left: the configuration stays put and the grid is rewritten. Right: the grid stays put and the configuration is sent somewhere else.

Lagrange’s equations do not care which reading one adopts for a mere change of coordinates. They are already covariant, so they keep the same geometric content after every passive transformation. They do care about active transformations. Only those active maps that leave L unchanged, or change it by a total time derivative, send motions to motions and produce conserved pairings of the canonical momentum with the generating vector field.

A practical test distinguishes the two. After a passive change, one is still looking at the original configuration, now written in new symbols. After an active change, one is looking at a different configuration, written in the old symbols. Cyclic coordinates, conserved impulse, and conserved angular momentum arise from the second situation, with an active family of displacements that L cannot detect. Changing from Cartesian labels to polar labels is the first situation with a passive rewriting that cannot, by itself, create or destroy a conservation law.

Figure 15.7: After a passive change one is still looking at the original configuration, now in new symbols. After an active change one is looking at a different configuration, written in the old symbols. Only the second can produce a conserved pairing.

Point Transformations

The objects that live naturally over configuration space are attached to points of Q , not to arbitrary mixtures of positions and velocities. It is therefore natural to change the description by moving, or relabeling, those points alone, and to let every velocity take care of itself.

Let φ be a diffeomorphism of Q , possibly depending on time, q'=φ(q,t). A curve of configurations is sent to a new curve q'(t)=φ(q(t),t). Differentiating that identity produces the only rule the velocities can obey

(15.33)

No independent decision is made about . The tangent lift is dragged along by the derivative of φ. When φ does not depend on t the extra term drops and the velocities transform by the ordinary Jacobian of a map of the base. A change of kinematic variables that is induced in this way by a (possibly time-dependent) map of configuration space is a point transformation.

Figure 15.8: A diffeomorphism φ of Q , possibly depending on time, sends a curve to a new curve. Differentiating that identity is the only rule can obey.

In the passive reading, φ is a change of charts, where the same configuration receives new labels, and the same tangent vector receives new components. In the active reading, φ is a genuine displacement of configurations—a translation, a rotation, or a more general rearrangement—and the tangent lift is the image lift. The formula is the same; the meaning is the distinction already drawn between relabeling and moving.

Because a point transformation is a map of the base, it acts automatically on every bundle built from T Q and . Vectors and one-forms at q are pushed or pulled by D φ. Components of the canonical momentum therefore adjust by the inverse Jacobian,

(15.34)

when the transformation is time-independent, so that the pairing remains the invariant it must be. The Lagrangian, being a scalar on T Q , is rewritten by substitution

(15.35)

If the original L did not depend on a particular combination of the old coordinates, the new L' will not depend on the image of that combination. Cyclic coordinates may appear or disappear from the component list; the underlying invariance of L under an active family of point transformations does not.

Time-dependent point transformations are allowed because time is not a coordinate on Q . They change what counts as velocity by the term and can turn an explicitly time-independent Lagrangian into an explicitly time-dependent one, or the reverse. They remain point transformations so long as the new configuration depends only on the old configuration and on t, never on .

What they are not is a general mixing of the slots in T Q . A rule that prescribed q' as a function of both q and would not come from a map of the base and would not send fibers of T Q→Q to fibers. Point transformations preserve the bundle projection, where they move, or relabel, configurations, and they carry each tangent space to the tangent space of the image point. That is why they are the changes of variable compatible from the outset with the geometry of motion in configuration space.

Figure 15.9: Because the map is of the base, it pushes vectors and pulls one-forms. The pairing is the invariant that forces the inverse Jacobian on p.

Noether's Theorem

A single active point transformation that leaves the Lagrangian unchanged sends motions to motions, but it does not yet produce a number constant along those motions. Constancy appears when the transformation belongs to a continuous family.

Let be a one-parameter family of point transformations with the identity, and let ξ be the vector field on Q that generates the family

(15.36)

Each acts on curves in the base and, by differentiation, on their tangent lifts.

Figure 15.10: is the identity. The generator ξ is the velocity of the family at s=0. Each drags curves and, by differentiation, their tangent lifts.

Suppose that, for every kinematic state,

(15.37)

or the same statement with . The action of every path then changes only by endpoint terms, so every motion is sent to a motion. Infinitesimally the condition is a single identity on T Q , where the first-order change of L along the lifted generator equals the time derivative of a function F of the configuration,

(15.38)

Along a motion Lagrange’s equations replace by . The identity collapses to

(15.39)

The pairing of the canonical momentum with the generator, corrected by the function that absorbed the possible total derivative, is therefore constant on every motion. That correspondence—continuous active point transformations that preserve L up to a total time derivative, producing an invariant constant along solutions—is Noether’s theorem.

The earlier special cases are this statement in adapted language. If and L does not depend on , then F=0 and the invariant is the cyclic momentum . If the family is spatial translation, the invariant is the corresponding component of the impulse. If the family is rotation about an axis, the invariant is the corresponding component of the angular momentum. In each case the level set of the invariant is an invariant submanifold of T Q on which the tangent lift is trapped.

The function F is not an ornament. Some symmetries change L by a divergence without changing the equations of motion. The conserved pairing must subtract that divergence, or constancy fails. When F=0 the charge is simply .

Time translation fits the same pattern only after time is treated as a parameter, not as a coordinate on Q . If L has no explicit time dependence, shifting the origin of t does not change the value of L at a kinematic state. The associated invariant is

(15.40)

again a scalar built from the pairing of momentum and velocity with L itself, and again constant on motions.

Passive transformations play no role in the theorem. Relabeling Q cannot create a charge. What creates a charge is an active family of point transformations that the variational data cannot detect. Noether’s theorem is the precise conversion of that undetectability into a first integral.

The Coordinate System-Independence of Noether’s First Integral

Noether’s construction produces a number at each kinematic state, with the pairing of the canonical momentum with the generating vector field, minus the function that accounts for a possible total derivative,

(15.41)

Along a motion that number does not change. What still has to be checked is that the number itself does not depend on how Q is labeled.

A passive point transformation replaces one chart by another and leaves the configuration, the curve, and the tangent lift where they are. The generator ξ is a section of T Q . Its components pick up the Jacobian,

(15.42)

The canonical momentum is a one-form, a section of along the lift. Its components pick up the inverse Jacobian,

(15.43)

The contraction is therefore the same in both charts

(15.44)

The correction F is a function of the configuration, hence already a scalar. Subtracting it cannot reintroduce chart dependence. The combination I is an invariant on T Q , not a privileged component in a privileged atlas.

The same conclusion follows from the variational argument that produced I. The family is a family of maps of the base. The statement that L changes at most by a total time derivative is a statement about values of a scalar on T Q . The identity

(15.45)

on motions is the vanishing of the time derivative of that scalar. No chart enters the hypotheses; none can enter the conclusion.

What does depend on coordinates is the appearance of the integral. In an adapted chart one may have , or , or I equal to a single Cartesian component of the impulse. After a generic relabeling the same I is a linear combination of the new momentum components, with coefficients equal to the new components of ξ. The formula looks different; the level set

(15.46)

is the same subset of the tangent bundle, and it remains an invariant submanifold.

Thus a first integral supplied by Noether’s theorem is coordinate-system-independent in the only sense the geometry allows: it is a scalar function on T Q , constant along the tangent lift of every motion, and its level sets are intrinsically defined. Changing generalized coordinates rewrites the scalar; it does not replace it by another conservation law, and it does not destroy the one already in hand.

Action of the (Weak) Symmetries on the Solutions of the Euler-Lagrange Equations

A solution of the Euler–Lagrange equations is a curve q(t) in Q whose tangent lift makes the Euler–Lagrange one-form vanish. The set of all such curves is the object a symmetry is entitled to move.

Take a one-parameter family of active point transformations . Applied to a curve, the family produces a deformed curve

(15.47)

together with the corresponding tangent lift. If L is strictly unchanged by every , the action of equals the action of q. Stationary curves are sent to stationary curves. Every solution is carried to a solution, and the inverse transformation carries it back. The family therefore acts on the set of solutions as a flow.

The same conclusion holds under a milder hypothesis. It is enough that each change L by the time derivative of a function of the configuration,

(15.48)

The action then changes only by endpoint terms. The Euler–Lagrange one-form is unaffected, because a total derivative never contributes to the bulk variation. Families of this kind still send solutions to solutions. They are weak symmetries of the variational problem, with symmetries of the equations of motion, and of the action up to a boundary piece, rather than symmetries of the function L itself.

A still weaker notion is sometimes useful. One may be given a family that is not known to come from an identity on L, but that can be checked to map the zero set of the Euler–Lagrange one-form to itself. Such a family acts on solutions without, by itself, guaranteeing a Noether integral. The integral requires the variational identity on T Q , on or off the solutions. Action on the solution set is the broader property; conservation is the narrower one.

In all three readings the action is active and geometric. It does not depend on a chart. A passive relabeling rewrites the same solution in new symbols and does not produce a second solution. The map does, q(t) and are generally different curves in Q . If ξ generates and is the associated first integral, every curve in the family shares the same value of I. The flow of a weak symmetry therefore runs inside a single invariant submanifold , sliding solutions along that submanifold rather than lifting them to a different level of I.

The infinitesimal form is the linearized action on curves. Differentiating at s=0 yields the vector field ξ evaluated along the original solution. That vector field is an infinitesimal deformation of one solution into neighboring solutions. When the symmetry is weak in the variational sense, the deformation is tangent to the solution set and tangent to the level set of I.

Thus a (weak) symmetry is not merely a decoration of L. It is an operation on the space of histories that preserves the Euler–Lagrange condition, and—when the variational identity holds—preserves the Noether integral as well. The solutions are not isolated curves; they come in families generated by those operations.

Jacobi’s First Integral, Invariance Under “Temporal Displacements” and Conservation of the Mechanical Energy

Time is not a coordinate on Q . Shifting its origin is nevertheless an operation on histories: every curve q(t) is replaced by the reparametrized curve q(t−ε), this occupies the same configurations at later instants. Velocities are unchanged, because the shift does not depend on t. If the Lagrangian does not depend explicitly on time,

(15.49)

then the value of L at a kinematic state is the same no matter when that state is examined. The variational data cannot detect the origin of the clock. That property is invariance under temporal displacements.

The corresponding identity on T Q is elementary. Differentiating along an arbitrary curve and using only the chain rule gives

(15.50)

once . On a motion the Euler–Lagrange equations replace by , and the right-hand side becomes

(15.51)

Rearrangement produces

(15.52)

The scalar

(15.53)

is therefore constant along every solution. That conserved combination is Jacobi’s first integral. It is built only from the pairing of the canonical momentum with the velocity and from the scalar L, so it is chart-independent in the same sense as any other Noether integral. Its level sets are invariant submanifolds of T Q .

Jacobi’s integral is not automatically the mechanical energy. It becomes energy when the velocity dependence of L is of a definite algebraic type. Suppose

(15.54)

where T is homogeneous of degree two in the velocities,

(15.55)

Euler’s theorem on homogeneous functions then supplies

(15.56)

and Jacobi’s integral collapses to

(15.57)

The right-hand side is the mechanical energy. Invariance under temporal displacements, together with that homogeneity, is conservation of mechanical energy.

If T is not homogeneous of degree two, or if V depends on the velocities, remains conserved whenever , but it need not equal T+V. If L depends explicitly on time—because a constraint is being moved from outside, or because a time-dependent point transformation has been used to rewrite the problem—temporal displacements are no longer symmetries and Jacobi’s integral is no longer constant. Energy bookkeeping then requires an account of the work done by the time-dependent data, which lies outside the closed variational problem on Q .

Passive changes of generalized coordinates cannot create or destroy the integral. They only rewrite in new components. What creates the integral is the active inability of L to notice a shift of the time origin. Noether’s theorem, applied to that inability, is Jacobi’s first integral. When the kinetic term is quadratic and homogeneous in the velocities, that integral is the mechanical energy.

Symmetries and Constants of Motion

A motion is a curve in Q . Anything that can be evaluated on its tangent lift and that returns the same number at every instant is a restriction the motion cannot escape. A real function C:T Q→R with

(15.58)

along every solution of the Euler–Lagrange equations is a constant of the motion. Equivalently, every tangent lift that meets a level set of C remains in that level set, so each regular level set is an invariant submanifold of T Q .

Some constants of the motion are accidents of a particular chart. Most of the useful ones are not. They come from operations that the variational data cannot detect. A continuous family of active point transformations that leaves L invariant, or changes L only by a total time derivative, sends solutions to solutions and, by Noether’s theorem, supplies a scalar

(15.59)

constant along those solutions. Cyclic momenta, components of the impulse, components of the angular momentum, and Jacobi’s first integral are this construction in special cases, where the generator ξ is a coordinate direction, a translation, a rotation, or the displacement of the time origin, and C is the associated pairing, possibly corrected by F or written as .

The correspondence runs in one direction more cleanly than in the other. Every (weak) point symmetry of the kind just described produces a constant of the motion. A constant of the motion need not, by itself, display the family that gave rise to it. In an unadapted chart the same scalar is a combination of several momentum components, and the generating vector field is not obvious from the formula. The geometry is still the same, where somewhere on Q there is an active displacement that L treats as invisible, and C is the invariant manufactured from that invisibility.

Passive transformations do not belong to the correspondence. They rewrite components; they do not produce new constants and they do not destroy old ones. After a change of generalized coordinates a constant of the motion remains the same function on T Q , expressed in new symbols. That is why one may hunt for symmetries in whatever chart makes the generator look simple, then evaluate the resulting C wherever convenient.

Several constants of the motion may be held at once. If are independent, the simultaneous level set

(15.60)

is again invariant. The tangent lift of every motion is confined to one such leaf, fixed by the values the constants take at the initial instant. Reduction consists in searching for solutions inside that leaf rather than in the whole of T Q .

The mechanical content of the section is therefore a dictionary. An active continuous symmetry of the Lagrangian on T Q is an operation on histories. A constant of the motion is a scalar on T Q that every history in the operated family must share. Noether’s theorem translates one into the other. The translations, rotations, ignorable coordinates, and temporal displacements already studied are the entries one meets first; the dictionary is not limited to those entries.

General and Global Formulation of Noether’s Theorem

The special cases already in hand—cyclic coordinates, translations, rotations, temporal displacements—share one structure. A continuous family of operations on histories changes the Lagrangian by a total time derivative, and a scalar built from the canonical momentum, the generator of the family, and that derivative is constant on solutions. What remains is to write the structure without a preferred chart, without separating “space” from “time” by hand, and without assuming the generator is independent of t.

The arena is still the tangent bundle. A generator is a rule that deforms curves in Q and, if desired, the time parameter along those curves. Infinitesimally one specifies a vector field along histories,

(15.61)

together with a coefficient τ(q,t) for a displacement of the time origin. The pair (ξ,τ) produces the deformed history

(15.62)

to first order in ε. Velocities change by the prolonged rule fixed by differentiation; no further data are chosen. When τ=0 this is an ordinary, possibly time-dependent, point transformation. When ξ=0 and τ=1 it is a pure temporal displacement.

Suppose the variational data cannot detect the deformation except at the endpoints. That is where along every curve, not only along solutions,

(15.63)

for some function F of (q,t). The identity is an equality of scalars on the space of kinematic histories. It does not refer to a chart except as a means of writing components of the section ξ and of the one-form . A pair (ξ,τ) satisfying the identity is a weak symmetry in the general sense, with an infinitesimal operation that changes the action by a boundary term.

On a motion the Euler–Lagrange equations convert the identity into a pure time derivative. The scalar

(15.64)

then obeys

(15.65)

That statement—every weak symmetry (ξ,τ,F) produces a chart-independent first integral I, constant along solutions—is the general and global form of Noether’s theorem.

The earlier integrals are this one formula. A cyclic coordinate corresponds to τ=0, F=0, and , so . A spatial translation or rotation corresponds to τ=0, F=0, and ξ the appropriate generator on Q , so I=p(ξ). A pure temporal displacement with corresponds to ξ=0, τ=1, F=0, so

(15.66)

Jacobi’s first integral is Noether’s integral for time translation. When T is homogeneous of degree two and L=T−V, that integral is the mechanical energy, up to sign conventions in the placement of .

Globality is the insistence that none of the ingredients is a coordinate list. The generator ξ is a section of T Q along histories, or of the pull-back of T Q to the domain of the curve. The momentum p is a section of along the tangent lift. The pairing is the duality pairing. The functions L, F, and τ are scalars. The combination I is therefore a scalar on the space of kinematic states. Its level sets are intrinsically defined invariant submanifolds of T Q , independent of the atlas used to compute the components.

The identity that defines a weak symmetry is off-shell, thus it is imposed on L along every curve. Conservation is on-shell, where dI/dt=0 uses the Euler–Lagrange equations. The distinction is the same one that separates “the variational data cannot detect the family” from “the number I does not change along a motion.” Action of the family on the set of solutions follows from the off-shell identity; constancy of I follows once those solutions are substituted.

A passive change of generalized coordinates rewrites , , and the component form of the identity. It leaves (ξ,τ,F) and I where they are. Noether’s theorem, in this formulation, is already a statement about sections, pairings, and scalars. Coordinate formulae are optional transcriptions of that statement, not part of its content.

Session 2

The Action Principle in Minkowski Spacetime

Until now a motion has been a curve in a configuration space of instantaneous shapes, parametrized by a distinguished time. For a single structureless particle in special relativity there is no such split at the outset. The theater of possible events is Minkowski spacetime M, a four-dimensional affine space equipped with a translation-invariant pairing η of signature (−,+,+,+). A history is a curve in that theater,

(15.67)

and the parameter σ need not be anyone’s coordinate time. The configuration space of the particle is spacetime. The tangent lift is the four-velocity dx/dσ at each event on the curve.

Two nearby events on a timelike curve determine an interval

(15.68)

with τ the proper time of that segment. Adding the segments along a world-line between two fixed events produces a single number, the elapsed proper time. Among all timelike curves with those endpoints, the straight line—the inertial world-line—makes that number stationary; in Minkowski spacetime it is a maximum. The variational problem whose stationary paths are the free motions is therefore the problem of extremizing proper time.

To obtain an action with the usual dimensions one multiplies by −m c and writes

(15.69)

The integrand is a scalar on the tangent bundle T M. It depends on the event and the tangent vector only through the Minkowski pairing of that vector with itself, so it is invariant under every active Poincaré transformation of M be it translations, rotations, and boosts. Those are point transformations of the configuration space. The overall minus sign is a convention that makes S smaller for longer proper times, in keeping with the habit of speaking of a stationary action rather than a maximal clock reading.

The same integrand does not depend on how the world-line is parametrized. If σ is replaced by , with , the factor under the square root and the measure dσ compensate, and S is unchanged. Coordinate time is one allowed parameter, not a privileged one. The Euler–Lagrange equations obtained from S are consequently reparametrization-invariant, where they constrain the image of the curve in M, not the speed at which the parameter runs along that image.

In a Lorentz frame one may choose σ=t and expand the square root for ∣v∣≪c. Up to a constant that does not affect the variation,

(15.70)

and is the free non-relativistic action on ordinary configuration space. The Newtonian problem is the low-velocity transcription of the proper-time problem. The geometry that was hidden in that transcription is the Minkowski pairing on T M.

Because S is a Poincaré scalar, Noether’s theorem applies directly on M. Invariance under translations supplies a constant pairing of the canonical momentum

(15.71)

with the translation generators—the four-momentum of the particle. Invariance under rotations and boosts supplies the conserved angular-momentum tensor about a chosen origin. The statements are the same ones already met in Newtonian language, now written as pairings on along a world-line.

The principle has not changed. One still compares nearby curves in the configuration space and demands that the action be stationary. What has changed is the configuration space itself, where it is now the spacetime of events, the Lagrangian is (minus a factor times) the Minkowski length of the tangent vector, and the physical path is a world-line of stationary proper time.

Hamiltonian Mechanics

Gradient Systems

A function on the space of states can generate a motion in more than one way. The most naive way is to let the system slide so as to decrease that function as fast as the available pairing allows.

Let X be the space where the instantaneous state is recorded—ordinary Euclidean space will do—and let V:X→R be a smooth scalar. At each point the differential dV is a one-form that eats a tangent vector and returns the corresponding directional derivative of V. To obtain a vector field one must turn that one-form into a vector. A constant positive-definite pairing ⟨⋅,⋅⟩ on X supplies the identification, where there is a unique vector ∇V such that

(15.72)

for every w. The ordinary differential equation

(15.73)

then points, at each state, in the direction of steepest descent of V. A dynamical system of this form is a gradient system.

(15.74)

The function V never increases. Isolated minima of V are asymptotically attracting; maxima are unstable. Closed orbits are impossible wherever the pairing is positive-definite, because a periodic solution would force V to return to its original value after having strictly decreased wherever ∇V≠0.

In coordinates orthonormal for the pairing the equation is elementary,

(15.75)

and the right-hand side is minus the list of partial derivatives. That list transforms as a one-form. It becomes a legitimate vector field only after the pairing has raised the index. The geometry of a gradient system is therefore the geometry of a scalar together with a pairing that identifies one-forms with vectors.

None of this is Hamiltonian mechanics. It is the comparison case. A single function is again about to generate a vector field, and the states will again move along the integral curves of that field. What will change is the rule that converts the differential of the function into a vector, this is not a positive-definite pairing, but the pairing supplied by the cotangent-bundle structure already used to define the canonical momentum. The consequences—conservation rather than dissipation, oscillations rather than collapse to a minimum, a different transformation law for the equations—are the reason for introducing gradient systems first. They show what a “system derived from a function” looks like when the conversion rule is the most obvious one, before that rule is replaced.

Phase Space

A point of configuration space names a shape, not a state of motion. The Lagrangian description repairs the omission by adding a velocity, where the instantaneous kinematic data are a point of T Q . The canonical momentum offers a different completion. At each configuration (q) one records, instead of , the one-form

(15.76)

already obtained by differentiating L with respect to the velocity. The pair (q,p) specifies the configuration together with the momentum that the variational principle attaches to it.

The set of all such pairs is the total space of the cotangent bundle . Its points are not velocities. They are configurations equipped with one-forms, and the bundle projection

(15.77)

simply forgets the one-form. A curve in is a history of both position and momentum; its projection is a curve in configuration space. When the fiber map coming from L can be inverted, every tangent lift in T Q corresponds to exactly one such curve, and the two descriptions record the same motion in different variables.

The space in which those pairs (q,p) live—the cotangent bundle of configuration space, taken as the arena of the first-order dynamics generated by the Hamiltonian—is phase space.

A chart on Q supplies, as before, the coordinate basis of and the dual basis of . The components of the momentum in that dual basis, together with the themselves, are coordinates on phase space. Under a time-independent point transformation of Q the change by the map of the base and the change by the inverse Jacobian, so that the pairing remains a scalar. Phase-space coordinates of this kind are those adapted to the bundle structure. They are not the only coordinates one may write on , but they are the ones in which the passage from L to

(15.78)

is transcribed most directly.

Gradient systems needed a single space of states and a pairing that turned dV into a vector. Phase space is the space of states appropriate to the Hamiltonian rule. The pairing that will turn dH into a vector field is not a positive-definite inner product on . It is the pairing of the base directions with the fiber directions that the cotangent bundle carries by construction, where displacements of q against displacements of p. The geometry of that pairing is the next subject. The present one is only the identification of the arena, where motions will now be curves in , and the variational data, once the velocities have been eliminated, will be the single function H on that space.

The Legendre Transform

The exchange of one set of independent variables for the dual set, by subtracting a pairing from a scalar, is not new in this course. It appeared first in Lesson 9, where the thermodynamic potentials are Legendre transforms of one another. Internal energy U(S,V) treats entropy and volume as the natural variables. Enthalpy, Helmholtz free energy, and Gibbs free energy are obtained by subtracting the pairings P V or T S, so that the natural variables become (S,P), (T,V), or (T,P). The numerical value of the potential changes; the underlying equilibrium data do not. What changes is which quantity is held fixed and which is allowed to vary.

The same construction now appears on the bundles over configuration space. The Lagrangian is a function of position and velocity. The Hamiltonian is to be a function of position and momentum. Passing from one to the other is not a change of chart on a single space. It is a fiber-wise exchange of the tangent and cotangent data sitting over the same configuration, exactly as a thermodynamic potential exchanges an extensive variable for its conjugate intensive one.

At a fixed q in Q the restriction of L to the fiber is an ordinary function of the velocity,

(15.79)

Its differential in the fiber is the canonical momentum already obtained,

(15.80)

That assignment

(15.81)

is the fiber derivative, or Legendre map. It covers the identity on the base, where the configuration is not moved. Only the extra datum in the fiber is replaced by its dual. In the language of Lesson 9, plays the role of the variable being transformed away, and p is the conjugate variable that replaces it.

The Legendre transform of is the function on the image of that map defined by

(15.82)

where is any velocity that produces the given p. Assembling these fiber-wise transforms gives a function on (an open set of) phase space,

(15.83)

The pairing is the duality pairing of with . It does not depend on a chart. Neither does L. Hence H is a scalar on whenever it is defined. The minus sign in the thermodynamic potentials and the arrangement here are the same device: subtract the pairing so that the differential of the new function involves dp rather than .

For the construction to yield a single-valued function of p, the fiber derivative must be invertible in the velocity. That is a condition on the Hessian

(15.84)

If W is non-degenerate, 𝔽 L is a local diffeomorphism of fibers, velocities can be solved for as , and H is well-defined in a neighborhood of the corresponding points of . A Lagrangian with this property is regular. If in addition W is positive-definite, the fiber maps are globally invertible on each tangent space for the usual mechanical Lagrangians, and the image of 𝔽 L is all of . Thermodynamics made the same demand, that a potential can be Legendre-transformed in a given slot only when the conjugate variable is a strictly monotone function of the original one, so that the map can be inverted.

When L is not regular the Legendre map fails to be invertible. Its image is a proper subset of phase space, the momenta are not independent, and constraints appear. That is the starting point of the Dirac analysis. The present subsection assumes regularity so that every kinematic state corresponds to exactly one phase-space point (q,p) and conversely.

The transform is involutive on regular Lagrangians, as it was on the thermodynamic potentials. Starting from H one recovers the velocities by

(15.85)

and the original Lagrangian by the same pairing,

(15.86)

In that sense L and H are dual presentations of one variational datum, one living on T Q and the other on . Performing the transform twice returns the original function, just as passing from U to F and back recovers U.

The algebraic type of L determines the meaning of H. If

(15.87)

with T homogeneous of degree two in the velocities, Euler’s theorem gives , and therefore

(15.88)

The Hamiltonian is then the mechanical energy, now written as a function of q and p rather than of q and . If T is not homogeneous of that degree, or if the potential depends on velocity, H remains the Legendre transform and remains a scalar on phase space, but it need not equal T + V. Jacobi’s integral on T Q is the pull-back of this H by the Legendre map whenever L has no explicit time dependence.

A time-independent point transformation of Q acts on T Q and on simultaneously. Velocities transform as tangent vectors, momenta as one-forms, and both L and the pairing are invariant. The Legendre transform commutes with the point transformation, where the Hamiltonian computed in the new chart is the old Hamiltonian expressed in the new coordinates. Passive relabeling cannot create a Hamiltonian, and it cannot destroy one. What can fail is regularity. A change of generalized coordinates may make the Hessian look different, where it cannot change whether the fiber derivative is locally invertible, because that is a property of the map of bundles.

If L depends explicitly on time, the same formulas define H(q,p,t). Time is a parameter, not a coordinate on Q . The Legendre transform is still performed in the fibers, instant by instant. Explicit time dependence is inherited, not manufactured.

The thermodynamic analogy should not be pressed into identity. In Lesson 9 the transform traded equilibrium variables inside a space of states that was not itself a cotangent bundle, and convexity guaranteed stability of equilibria. Here the transform trades velocity for momentum inside the fibers of T Q → Q , and non-degeneracy of the Hessian guarantees that phase space is a faithful copy of the kinematic states. The algebra is the same; the geometry that carries the algebra is the bundle geometry assembled in this lesson.

Thus the Legendre transform is the precise dictionary between the two bundles over configuration space. It sends a regular Lagrangian on T Q to a Hamiltonian on , it identifies the tangent lift of a motion with a curve in phase space, and it prepares the first-order equations that treat q and p as equal partners. Those equations are next.

Hamilton’s Equations

The Lagrangian description produces a second-order system on Q . Once the Legendre map has replaced velocities by momenta, the same motion is a first-order system on . A single function on phase space is enough to write that system, provided one knows how to turn the differential of the function into a vector field.

With the velocities eliminated by the inverse of 𝔽L,

(15.89)

the Hamiltonian of the previous subsection is a scalar on phase space. Differentiating its definition and using the inverse Legendre map yields

(15.90)

Along a motion the Euler–Lagrange equation replaces the second right-hand side by . The result is Hamilton’s equations

(15.91)

They are first-order. Position and momentum enter symmetrically, except for a sign. The curve they determine in projects to a curve in Q whose tangent lift recovers the original Lagrangian motion whenever the Legendre map is invertible.

Figure 15.11: Once the inverse Legendre map has eliminated velocities, one function H writes the whole system. Position and momentum enter symmetrically, except for a sign..

Suppose that, for every kinematic state,

The same equations have a geometric reading that does not begin with components. The differential dH is a one-form on phase space. The cotangent bundle carries a pairing that identifies one-forms with vector fields: base displacements δ q against fiber displacements δ p. The unique vector field satisfying that identification is the Hamiltonian vector field of H. In adapted coordinates

(15.92)

Hamilton’s equations say that the phase-space curve is an integral curve of .

(15.93)

Time evolution is the flow of that vector field.

Figure 15.12: The canonical pairing identifies the one-form dH with a unique vector field . Hamilton’s equations say the phase-space curve is an integral curve of that field. Time evolution is its flow.

The contrast with a gradient system is now sharp. There a positive-definite pairing turned dV into −∇V, and V decreased along the motion. Here the pairing is the canonical pairing of q-directions with p-directions. It is antisymmetric. The same function H therefore generates a circulation in phase space rather than a collapse to a minimum. In particular

(15.94)

whenever H has no explicit time dependence. The Hamiltonian is constant on its own flow. When the Legendre transform of a standard mechanical Lagrangian has been performed, that constant is the mechanical energy. Jacobi’s integral on T Q has become H on .

Covariance is a statement about the vector field, not about 2 n unrelated scalar equations. Under a time-independent point transformation of Q, the new coordinates (q',p') remain adapted to the bundle . The components of change by the Jacobian of that transformation on phase space, and Hamilton’s equations in the new chart are the component list of the same geometric equation . A generic mixing of all 2 n phase-space coordinates will destroy the simple split into and ; it does not destroy the vector field. What is privileged is the pairing that defines , not a particular list of symbols.

Figure 15.13: A gradient system uses a positive-definite pairing and decreases V. The canonical pairing is antisymmetric, so H generates circulation and is constant on its own flow. That constant is the mechanical energy when the Legendre transform of a standard Lagrangian has been performed.

If L depends explicitly on time, so does H. Hamilton’s equations keep the same form as (15.92), but now

(15.95)

Temporal displacements are no longer symmetries, and energy bookkeeping must include the explicit time dependence. The vector field is still well-defined at each instant; the flow may simply fail to preserve the value of H.

The variational origin is not lost. The phase-space action

(15.96)

has first variation vanishing, for fixed endpoints in q, if and only if Hamilton’s equations hold. The boundary term involves only δ q, that is why p may be varied freely. In this form the stationary principle lives directly on . The Lagrangian principle on T Q is recovered by substituting the inverse Legendre map.

Thus Hamilton’s equations are three things at once. The first-order transcription of Euler–Lagrange after the velocities have been eliminated. The integral curves of the vector field manufactured from dH by the canonical pairing. The stationarity condition for the phase-space action. The next task is to name that pairing as a two-form and to read off the geometric properties of the flow it generates.

Harmonic Oscillators

The first system that Hamilton’s equations should be read for is the one whose motions are already known. A harmonic oscillator is a single degree of freedom whose configuration space is a line, Q = ℝ, with coordinate q, and whose Lagrangian is

(15.97)

with m > 0 and k > 0. The first term is kinetic energy, homogeneous of degree two in the velocity. The second is a potential with a nondegenerate minimum at the origin. Nothing in L depends on time, and q is not cyclic.

The fiber derivative is elementary. Holding q fixed and differentiating with respect to velocity gives the canonical momentum

(15.98)

The Hessian is the number m, and is non-degenerate, so the Lagrangian is regular. The inverse Legendre map is  , and the Hamiltonian is

(15.99)

This is T + V, as the general rule for a quadratic kinetic term already promised. Phase space is the plane with coordinates (q,p). The level sets of H are ellipses

(15.100)

provided E > 0. The origin is the single critical point, corresponding to the rest configuration at the bottom of the well.

Hamilton’s equations are

(15.101)

They are first-order and linear. Differentiating the first and substituting the second recovers the familiar second-order equation

(15.102)

on configuration space. The Hamiltonian vector field is

(15.103)

Its integral curves wind around the origin. They do not run down the gradient of H. A gradient system for the same function H would point toward the origin and stop there. The canonical pairing instead sends dH to a vector tangent to the ellipses, so the motion stays on a single level set. That is Liouville’s theorem in one degree of freedom, the flow slides along the energy curves and does not crush area.

Figure 15.14:  From the line to the energy ellipses. Configuration space is a line. The fiber derivative opens the cotangent plane. Level sets of H = T+V are ellipses about the single rest point at the origin.

The angular frequency

(15.104)

converts the ellipse into uniform circular motion after a linear change of scale. The general solution

(15.105)

is a periodic orbit of period 2π/ω. The amplitude A and the phase φ are constants of the motion in the elementary sense; equivalently, they label which ellipse is occupied and where on that ellipse the initial data sit. Energy

(15.106)

is the value of H on that ellipse. Because H has no explicit time dependence, E is constant, as Jacobi’s integral already required on T Q.

A chart change q |→ q' = λ q, with λ a nonzero constant, rescales the coordinate and the conjugate momentum by inverse factors. The ellipse is rewritten; the orbit in is the same set of states. Passive relabeling cannot turn an oscillator into a different mechanical system. An active translation q |→ q + a does change L, unless k = 0, so there is no conserved linear momentum. The only continuous point symmetry of this L is time translation, and its Noether charge is H itself.

Several uncoupled oscillators are the same construction on a product configuration space. If

(15.107)

the Legendre map acts factor by factor, H is a sum of independent elliptic Hamiltonians, and phase space is a product of planes. Each factor contributes its own periodic orbit. The combined motion is periodic if the frequencies are commensurate, and densely fills a torus in the product of ellipses if they are not. That picture—independent harmonic motions on a torus in phase space—is the linear model of every later integrable system.

A driven or damped oscillator is a different object. Explicit time dependence in L, or a velocity term that is not a total derivative, takes the problem outside the closed variational setting used here. The Hamiltonian may still be written at each instant, but it need not be constant, and the flow need not preserve the ellipses of the undriven H. Those cases belong with time-dependent Hamiltonians and with dissipative vector fields, not with the example that exhibits Hamilton’s equations in their cleanest form.

Thus the harmonic oscillator is the dictionary entry that should be kept in hand. Configuration space is a line. Phase space is a plane. The Legendre transform turns a quadratic Lagrangian into an elliptic Hamiltonian. Hamilton’s equations are linear and periodic. The flow of is circulation on energy ellipses, not descent to a minimum. Every later claim about symplectic geometry, Liouville flow, and small oscillations around an equilibrium is this example, rewritten in a more general chart.

Global Solutions to Hamilton’s Equations

Hamilton’s equations are an ordinary differential equation on phase space. A choice of initial point in determines, at least for a short time, a unique curve

(15.108)

with and . That local statement is the standard existence and uniqueness theorem for a smooth vector field. What the theorem does not grant, by itself, is that the curve can be continued for all time.

A solution defined on a maximal open interval containing the initial instant is a maximal integral curve of . If is finite, the curve leaves every compact subset of as t approaches , where either some coordinate diverges, or the representative in a given chart ceases to make sense because the motion has left the chart. The same holds as t approaches . A Hamiltonian vector field for which every maximal integral curve has and is complete. Its flow is then a one-parameter group of diffeomorphisms of phase space,

(15.109)

defined for every real t. Completeness is a global property of , not a local property of Hamilton’s equations in a chart.

Conservation of H is the first obstruction to blow-up, and the first aid to completeness. If H has no explicit time dependence, every integral curve is trapped in a single level set

(15.110)

If that level set is compact, the curve cannot escape to infinity in finite time, and the solution exists for all t. Compact energy surfaces are therefore a geometric sufficient condition for global existence. The harmonic oscillator is the model case where each ellipse is compact, the flow is periodic, and is a rotation of that ellipse defined for all time.

A proper Hamiltonian with a lower bound gives the same conclusion without naming ellipses. Suppose H is smooth, bounded from below, and proper as a map , so that each sublevel set H ≤ E is compact. Then every energy surface that appears in the dynamics is compact, is complete, and the flow exists globally. Ordinary mechanical systems

(15.111)

with kinetic energy positive-definite and quadratic in p, and with V smooth and proper on Q , fall under this heading. The particle cannot reach spatial infinity in finite time without sending H to infinity, which the equations forbid.

The condition is not automatic. A smooth Hamiltonian on a noncompact phase space may have solutions that escape in finite time. If V decreases too steeply, or if H is polynomial of the wrong degree, the velocity can grow fast enough that q(t) diverges at a finite . The vector field remains smooth where it is defined; the integral curve simply leaves every compact set. In that case there is no global flow on all of , only a local flow on a t-dependent domain. Completeness can also fail for an otherwise innocent H written in a chart that covers only part of the true phase space, where the solution may be global on the manifold and look singular only because the coordinates have broken down.

Time-dependent Hamiltonians change the bookkeeping, not the local ODE theorem. One still has unique maximal integral curves, but H is no longer constant along them, and compactness of the instantaneous level sets of H(·,t) need not prevent escape. A driven oscillator with a smooth bounded drive remains complete; a drive that injects energy without bound can send the representative out of every compact set, even if each frozen Hamiltonian H(·,t₀) would have been complete by itself.

Reparametrization-invariant systems, such as the free relativistic particle with the square-root action, require a separate sentence. The Euler–Lagrange equation constrains the image of the world-line, not the speed of the parameter. Completeness is then a statement about the world-line in Minkowski spacetime, not about a particular choice of σ. After a Hamiltonian reduction that isolates proper time or coordinate time as the parameter, the question becomes again whether the reduced vector field on the reduced phase space is complete.

A global solution is therefore not merely a formula that happens to be written without restriction on t. It is an integral curve of that exists for all time as a map into . When every such curve is global, the dynamics is a flow, the flow preserves the symplectic form that will be introduced soon, and the language of one-parameter groups already used for symmetries of Q applies to time evolution itself. When some curves are not global, the geometry is still local and Hamiltonian; one simply does not have a single group of diffeomorphisms of the whole phase space.

Liouville’s Theorem

A motion in phase space is a curve. A cloud of motions is a region. Take a bounded set of initial data in , a collection of points (q, p), and let each point run under Hamilton’s equations. At a later time the same points occupy a new set. Liouville’s theorem says that the new set has the same phase-space volume as the old one. Points are neither created nor destroyed, and they are not crowded into a smaller volume. The flow rearranges them; it does not compress them.

The counting is the geometric content. Phase space is the space of states. A point of is one complete instantaneous state, configuration together with momentum. A region U is a set of such states. The Hamiltonian flow sends each state to the state it becomes after time t, so is the same set of histories, read at a later instant. If the flow exists on the interval in question, the map is a diffeomorphism of its domain onto its image. Distinct initial points remain distinct. No two histories merge, and no history appears that was not already present in U. Conservation of points means that is one-to-one on U and that every point of the image is the time-t state of exactly one point of U.

Figure 15.15:  A region of states, read at two times. The left panel is a bounded set of initial data. The right panel is the same set of histories at a later instant, . The map is a diffeomorphism of its domain onto its image: no two points merge, and no point of the image lacks a preimage in U.

Volume is the measure of how much of phase space those points occupy. In adapted coordinates it is the ordinary 2n-dimensional volume

(15.112)

Liouville’s theorem is the equality

(15.113)

for every region U where the flow is defined, and for every t in that interval. The cloud may stretch in the q-directions and thin in the p-directions, as it does for a free particle, or it may rotate, as it does for the harmonic oscillator. The product of the stretchings is one. A density of points, constant on U at t = 0, remains constant along the motion. The representative points of an ensemble do not accumulate and do not rarefy.

The local form is a statement about the vector field. Let ρ be a density of states, a function on phase space whose integral over a region is the amount of that region occupied by the cloud. Transporting the cloud with gives

(15.114)

Hamilton’s equations force the divergence to vanish. In adapted coordinates

(15.115)

because mixed second partials of H agree. The continuity equation therefore reduces to

(15.116)

along the flow. A density that starts uniform stays uniform. That is incompressible flow on phase space, a direct analogy to the same statement one makes for a fluid whose velocity field has vanishing divergence, now made for the velocity field on .

Figure 15.16:  Why the divergence vanishes.

The contrast with a gradient system is again the point of the comparison. The vector field −∇V has divergence −Δ V, that need not vanish. A cloud of points sliding downhill can shrink onto a minimum. Phase-space volume is not conserved, and points that began separated can be packed into an arbitrarily small neighborhood of an attractor. Hamiltonian flow has no attractor of that kind. Equilibria are critical points of H, not sinks of volume. The harmonic oscillator makes the distinction visible where every energy ellipse keeps its area, and a small disk of initial data rotates and shears but returns with the same area.

The theorem does not say that shapes are preserved. A ball of initial data for a free particle becomes a sheared slab, long in q and thin in p, of unchanged volume. It does not say that H is the only constant of the motion. It says that the motion itself, as a rearrangement of points, is volume-preserving. When the flow is complete, is a one-parameter group of volume-preserving diffeomorphisms of . When the flow is only local, the same equality holds on every region that has not yet escaped.

Passive changes of adapted coordinates do not create the conservation. The coordinate volume dq dp transforms by the absolute value of the Jacobian. On the charts adapted to that Jacobian has absolute value one precisely when the change is symplectic, so the number Vol(U) is the same in every such chart. A generic mixing of all 2n coordinates can make the coordinate integral look different; the intrinsic volume defined by the pairing is unchanged. What is conserved is that intrinsic volume of a set of states, not a privileged coordinate expression for it.

As we shall see in a later chapter, statistical mechanics begins from this fact. An ensemble is a cloud of points in . Liouville’s theorem says the cloud is incompressible, so a density constant on the energy surface remains constant, and the only question left is which surface, and which distribution on it, the preparation of the system has selected. Numerical integrators that are themselves symplectic copy the same conservation at the discrete level, where they rearrange representative points without systematically creating or destroying phase-space volume. Integrators that do not will, over long times, spiral the cloud inward or outward and counterfeit a dissipation the equations do not contain.

Thus Liouville’s theorem is the global expression of the vanishing divergence of the Hamiltonian vector field. Points of phase space are conserved as identities of histories, and the volume they occupy is conserved as a number. The linear maps that preserve the pairing, and therefore this volume, are the symplectic group (we will se that a bit later).

The Wedge Product

The wedge product is the skew pairing that turns two vectors into an oriented area, and two one-forms into a two-form. It is the product used to write the symplectic form

A one-form eats one vector and returns a number. A two-form eats two vectors and returns a number, and it changes sign if the two vectors are swapped. The wedge product is the rule that builds the second object out of two copies of the first.

Given one-forms α and β, their wedge α ∧ β is the two-form defined on a pair of vectors u, v by

(15.117)

The minus sign is the whole point. It forces

(15.118)

Swapping the factors changes the sign. Feeding the same one-form in twice gives nothing, because a parallelogram spanned by one direction twice has no area.

On the coordinate one-forms of a chart this is elementary. The value

(15.119)

is +1 on the ordered pair , -1 on the swapped pair, and 0 if the two vectors are not that pair. So measures the signed area of the parallelogram spanned by the first two coordinate directions, in that order. Any two-form is a linear combination of such elementary wedges.

The same rule applies to vectors, with the arrows reversed. The wedge u∧v is the signed parallelogram they span, where swapping u and v reverses the orientation, and u∧u=0. A one-form paired with a vector is a number; a two-form paired with a wedge of two vectors is the signed area of that parallelogram as measured by the two-form.

We have already demonstrated the property of skew-symmetry in (15.119). We can also state that the wedge is linear in each factor. And that for a one-form and a two-form the sequence of factors matters, but once the order is fixed the product is unambiguous; we can call this a definite order property.

On phase space the adapted coordinates supply 2n one-forms. The canonical two-form is the sum of the area elements in each conjugate pair,

(15.120)

Evaluated on two displacements (δ q,δ p) and (δ q', δ p'),

(15.121)

That pairing is the one that turns dH into the Hamiltonian vector field . It measures signed area in each plane. It does not measure Euclidean length.

The top power is the volume. In two dimensions Ω itself is the area form. In 2n dimensions the n-fold wedge

(15.122)

is a nonzero multiple of

(15.123)

Liouville’s theorem is the statement that Hamiltonian flow preserves Ω, and therefore preserves this volume. The vanishing of the divergence of the vector field in adapted coordinates is the component form of that statement.

A passive change of adapted coordinates replaces and by new one-forms related by the Jacobian and its inverse. Because those transformations are inverses of each other, the sum is unchanged. The two-form does not depend on how the configurations are labeled. A generic mixing of all 2n phase-space coordinates produces a different two-form. The maps that leave Ω alone are the ones compatible with the pairing already used to write Hamilton’s equations.

Thus the wedge product is the skew multiplication that manufactures areas out of one-forms. The canonical two-form is the sum of the coordinate area elements . The conservation of phase-space volume is the conservation of the top power of that two-form.

Poisson Brackets

Hamilton’s equations answer a narrow question. Given H, how do the coordinates of one point move? Most of what we care about is not a coordinate. Energy is a function. So is angular momentum, a constraint, a Noether integral, the reading of an instrument. Each is a smooth real function f on phase space. The motion carries a point along the integral curve of , and f changes at the rate . That rate is already determined by H and f. It should not have to be recomputed from scratch every time the chart is changed.

The Poisson bracket is that rate, promoted from a derivative along one vector field to a bilinear operation on functions. Once it is in hand, the coordinates become dispensable. Two functions determine a third, and the third is the rate at which the first changes along the Hamiltonian flow of the second.

Figure 15.17:  The bracket is a rate read off level sets. is tangent to the levels of g. The directional derivative of f along that arrow is {f, g}.

Definition 15.1: Poisson Bracket: Let f and g be smooth real functions on an open set of , written in canonical coordinates . Their Poisson bracket is

(15.124)

with the summation convention on i = 1, …, n. The result is again a smooth function. It does not depend on a choice of curve. It depends on f, g, and the canonical pairing.

The same formula is the directional derivative along the Hamiltonian vector field of g. With

(15.125)

one has . In particular, along the actual motion,

(15.126)

The partial derivative keeps any explicit clock dependence. The bracket keeps everything the motion itself contributes. Hamilton’s equations are the special case in which f is a coordinate

(15.127)

So the bracket does not replace Hamilton’s equations. It is those equations, applied to an arbitrary observable.

The differential df is a one-form. The Hamiltonian vector field is the symplectic mate of dg. The bracket is the pairing of those two objects,

(15.128)

once the symplectic form is admitted. Nothing in the definition requires that language yet. What is required is the sign pattern already fixed by : raise the p-derivative into the q-slot, and insert a minus when the q-derivative is lowered into the p-slot. That minus is the same minus as in . Drop it, and the bracket no longer reproduces the motion.

Figure 15.18:  The fundamental bracket is the unit of area in a canonical chart. The Hamiltonian vector field is the gradient rotated so that it circulates, not descends.

Principle 15.1. Fundamental brackets. In canonical coordinates,

(15.129)

Every other bracket is built from these by the product rule. If a passive change of chart preserves these three relations, it preserves every bracket. That is the practical test that a chart is still canonical.

Principle 15.2. Algebraic properties. For smooth functions f, g, h and a constant c,

(15.130)

(15.131)

(15.132)

(15.133)

(15.134)

Antisymmetry (15.130) says that f is constant along the flow of f itself, when energy does not change along the energy flow. The Leibniz rule says the bracket is a derivation in each slot, as a directional derivative must be. The last line (15.134) is Jacobi’s identity. It is the statement that Hamiltonian flows close among themselves, so that the commutator of and is the Hamiltonian vector field of the bracket,

(15.135)

A failure of Jacobi would mean that the operation of “flowing along a function” did not form a Lie algebra. The mixed partials that cancelled in Liouville’s divergence are the same mixed partials that make Jacobi hold.

Definition 15.2. Constant of the motion. A function I with no explicit time dependence is a constant of the motion when {I, H} = 0. Equivalently, , where every integral curve of lies in a level set of I. The levels of I are invariant submanifolds, not merely sets of the same volume.

This is the bracket form of the statement already proved for a Noether integral. If is built from a point transformation that leaves L invariant up to a total derivative, then {I, H} = 0 in every canonical chart. The bracket makes the coordinate independence obvious. One no longer has to push the vector ξ through a new chart and recompute. Either the bracket vanishes, or it does not.

For example, for a free particle on the line. . For the momentum, {p, H} = 0, so p is constant. For the initial-position combination I = q − p t/m, the explicit time derivative cancels {I, H} = p/m, and dI/dt = 0. The bracket alone is not the whole story when the observable carries the clock.

For the harmonic oscillator. . Then {q, H} = p/m and , which is the linear system already solved. The energy satisfies {H, H} = 0. In the plane, the flow is circulation on the ellipses of H; the bracket of H with itself vanishes because is tangent to those ellipses.

Figure 15.19:  A constant of the motion is a function whose levels contain the flow. Vanishing of the bracket is tangency, not a numerical accident.

What the bracket does not do. It does not preserve the numerical value of a function; it produces the rate of change. It does not by itself choose which function is the Hamiltonian. It does not preserve shape of regions; Liouville already separated volume from shape. And a generic chart on the 2n-dimensional manifold, one that mixes q and p without protecting the pairing, will write a different coordinate formula. The operation on functions survives. The partial-derivative recipe is tied to canonical charts.

Two readings should be kept distinct. As an algebraic structure, the bracket turns the algebra of observables into a Lie algebra, with a derivation rule for products. As a geometric structure, it is the symplectic pairing of Hamiltonian vector fields. Canonical transformations are exactly the maps that preserve the bracket, hence preserve both readings. The linear ones, at a single point, are the symplectic group announced at the end of the volume argument.

Canonical Transformations

Phase space has many coordinate systems. Most of them are merely labels. Hamilton’s equations, the fundamental brackets, and Liouville’s volume were all written in charts adapted to the pairing of . A passive change that ignores the pairing will scramble into something that is no longer a Hamiltonian vector field. The new equations may still be correct as a system of ordinary differential equations. They will not be Hamilton’s equations for any function of the new coordinates.

A change of chart that protects the pairing is what we call a canonical transformation. Equivalently, it is a diffeomorphism of phase space that preserves the Poisson bracket. The two descriptions are the same requirement, read once in coordinates and once on functions. Point transformations of Q , lifted in the way already fixed by the fiber, are the special case in which the base is transformed and the fiber is carried along. Canonical transformations are wider, where they may mix coordinates and momenta.

Figure 15.20: What is being preserved. The left chart is canonical. The right chart is another labeling of the same open set. The transformation is canonical when the unit cell of the (q, p) pairing is sent to a unit cell of the (Q, P) pairing, not merely to a region of area one.

Definition 15.3. Canonical transformation. Let be canonical coordinates on an open set of , and let

(15.136)

be a smooth change of chart, possibly depending on the parameter t. The change is canonical when the new coordinates obey the fundamental brackets computed in the old chart,

(15.137)

The bracket on the left is the old Poisson bracket. The test does not require the new Hamiltonian to be known in advance. If these three relations hold, every bracket of functions is preserved, and Hamilton’s equations in the new chart are Hamilton’s equations for a function K(Q, P, t).

Principle 15.3. Preservation of the motion. If the transformation is canonical, there is a Hamiltonian K such that

(15.138)

When the transformation does not depend on t, one may take K(Q, P) = H(q(Q, P), p(Q, P)). The value of the energy is merely re-expressed. When the transformation depends on t, K differs from the re-expressed H by a term coming from that explicit dependence. The form of the equations is the invariant. The particular function K is not.

Principle 15.4. The symplectic condition. Write z = (q, p) and Z = (Q, P), and let M be the Jacobian matrix . Let J be the block matrix

(15.139)

with I the n × n identity. The transformation is canonical if and only if

(15.140)

The left-hand side is the pairing pushed forward. The right-hand side is the pairing in the new chart. Taking determinants gives , so det M = ±1. Orientation-preserving canonical transformations have det M = 1, and therefore preserve Liouville volume. Volume preservation is necessary but not sufficient for a map may keep areas and still fail to keep {Q, P} = 1. The bracket test is the sharper one.

Active and passive, again. Read passively, a canonical transformation is a new chart on the same set of states. The point of phase space does not move; its labels do. Read actively, it is a diffeomorphism sending each state to another state, carrying the Hamiltonian vector field along. Both readings use the same formulas. The passive reading is the one needed to simplify a problem. The active reading is the one needed when the map itself is a symmetry, generated by a constant of the motion.

Figure 15.21: The lift of a point transformation is the model case. A diffeomorphism of Q acts on configurations. Its lift to acts on the fiber by the inverse transpose, so that p dq is unchanged as a one-form. The lifted map is canonical, and it does not mix the base with the fiber.

We now turn, briefly, to generating functions. Locally, a canonical transformation that keeps q and Q independent as coordinates is the gradient of a single function

(15.141)

and, when depends on t,

(15.142)

Three other types trade which variables are independent, by Legendre transform of . The generating function is a convenience, not a definition. The definition remains the preservation of the bracket. Where a generating function exists, the symplectic condition is automatic, because equality of mixed partials is built into and .

If F carries t, the new Hamiltonian is not the old one rewritten. The added term is the price of using a moving chart. This is the same phenomenon already met when a constant of the motion was allowed to depend explicitly on t, where the total derivative along the motion, not the bracket alone, is the quantity that vanishes. A time-dependent canonical transformation can be used to remove the time from H entirely, when the problem permits it. The free particle and the oscillator both allow such charts; a generic V(q, t) does not.

Canonical transformations do not preserve H, only the Hamiltonian form of the equations. They do not preserve the splitting into base and fiber, except in the special case of extended point transformations. They do not have to arise from a symmetry. And the test {Q, P} = 1 must be computed with the old bracket; replacing the partial derivatives by derivatives with respect to Q and P assumes the conclusion.

Figure 15.22: The oscillator chart. Ellipses of H become horizontal lines of constant P. The flow, which circulated, becomes a uniform drift in Q at fixed P. Nothing physical moved. The chart was bent until the motion looked like a free particle in one coordinate.

The Symplectic Group

A linear change of coordinates on phase space is not automatically a change that Hamilton’s equations will accept. Phase space is even-dimensional. In adapted coordinates a point is a column of 2n numbers, the stacked above the . A constant invertible matrix S sends one such column to another. Most such matrices scramble the pairing of base directions against fiber directions. The ones that do not form a group.

Write the standard pairing in matrix language. Let J be the 2n by 2n block matrix that sends (δ q, δ p) to (δ p, −δ q). In one degree of freedom it is the quarter-turn

(15.143)

In n degrees of freedom the same pattern is repeated on the diagonal blocks. The bilinear form

(15.144)

is skew, nondegenerate, and exactly the pairing that turned dH into . A linear map S preserves that pairing if and only if

(15.145)

The set of all real 2n by 2n matrices satisfying this identity is the symplectic group Sp(2n, ℝ). It is a group where the identity satisfies the identity, the product of two solutions is a solution, and every solution is invertible with inverse , which is again symplectic. It is not the orthogonal group. An orthogonal matrix preserves a positive-definite inner product and therefore preserves lengths and angles. A symplectic matrix preserves an area form. It may stretch one direction and compress the dual direction, so long as the product of the two scalings is one.

The determinant of every symplectic matrix is +1. Preservation of ω implies preservation of the top power , which is a constant multiple of the Euclidean volume form on . Linear canonical transformations therefore preserve phase-space volume. That is the linear case of Liouville’s theorem, visible before any flow has been written down.

The Lie algebra is read off by differentiating the defining identity at the identity matrix. A curve S(ε) in Sp(2n, ℝ) with S(0) = I has derivative A = S'(0) obeying

(15.146)

or equivalently J A symmetric. Matrices with that property form the symplectic Lie algebra sp(2n, ℝ). Its dimension is n(2n + 1), the same as the dimension of the group. Every such A is the Hamiltonian vector field of a quadratic function. If H is a homogeneous quadratic polynomial on phase space, the matrix of lies in sp(2n, ℝ), and every element of the algebra arises in this way. The harmonic oscillator is the model entry: on the (q, p) plane,

(15.147)

and the one-parameter group it generates is a curve in Sp(2, ℝ), a periodic family of area-preserving linear maps that rotate the energy ellipses into themselves.

Not every matrix in Sp(2n, ℝ) comes from a point transformation of Q. A point transformation of the base, time-independent, lifts to a linear map on the fibers of that mixes the q’s among themselves and transforms the p’s by the inverse Jacobian. Those lifts form a subgroup. The full symplectic group is larger. It allows maps that mix q with p, which no relabeling of configurations can do. The exchange q' = p, p' = −q is symplectic and has no interpretation as a change of generalized coordinates. It is an active linear canonical transformation of phase space.

Passive and active readings must be kept apart, as they were for point transformations of Q . A passive symplectic change replaces one adapted chart on by another and leaves the point of phase space where it is. An active symplectic map sends each phase-space point to a different point and carries curves to curves. Hamilton’s equations are covariant under the first reading, where the vector field is the same geometric object, rewritten in new components. They are invariant under the second reading only when the active map sends the Hamiltonian vector field of H to the Hamiltonian vector field of the transformed H. Linear symmetries of a quadratic Hamiltonian are precisely the elements of Sp(2n, ℝ) that commute with the matrix of .

The group is the linear skeleton of the canonical transformations that will be needed once the pairing is named as a two-form. A diffeomorphism of that pulls that two-form back to itself is a nonlinear canonical transformation; its derivative at each point is an element of Sp(2n, ℝ). The group is what the derivative is allowed to be. Small oscillations about an equilibrium linearize to a quadratic Hamiltonian, and the monodromy of that linear system is a curve in Sp(2n, ℝ). Stability is then a question about that curve, not about the orthogonal group of the Euclidean metric on phase space, which the dynamics does not preserve.

Thus Sp(2n, ℝ) is the group of linear maps that treat phase space as a symplectic vector space rather than as a Euclidean space of twice the dimension of Q . It preserves the pairing that manufactures , it preserves volume, and its Lie algebra is the set of linear Hamiltonian vector fields. The two-form whose preservation defines the group is the next object.

Symplectic Geometry

The symplectic condition was a statement about charts. It declared what linear substitutions keep the fundamental brackets. That test is local and algebraic. It does not yet say what is being preserved when no chart has been chosen, nor why the same minus sign appears in Hamilton’s equations, in the Poisson bracket, and in the Jacobian condition.

In a canonical chart the two-form is the sum of the elementary area elements,

(15.148)

A vector along and a vector along span a parallelogram. The form ω returns the signed area of that parallelogram, with the sign fixed by the order (q-leg, then p-leg). Two vectors that both lie along the base, or both along the fiber, span nothing that ω can see. This is the pairing, written so that the answer does not depend on a choice of function, and the recipe uses coordinates only as a convenience.

Figure 15.23: The elementary area. One leg along , one leg along , and ω returns +1. The opposite order returns −1. A parallelogram lying inside a single fiber, or inside a single constant-p section, is invisible to ω.

Definition 15.4. Symplectic form. A symplectic form on a manifold M of dimension 2n is a 2-form ω that is closed and nondegenerate. Closed means dω = 0. Nondegenerate means that if ω(u, v) = 0 for every vector v at a point, then u = 0 at that point. The pair (M, ω) is a symplectic manifold. Phase space , with the form written above in any canonical chart, is the example this course has been using all along.

Nondegeneracy is the geometric form of the statement that the fundamental brackets determine the coordinates. Given ω, every function f determines a unique vector field by the rule

(15.149)

In a canonical chart this reproduces

(15.150)

this is the Hamiltonian vector field already in hand. The minus sign is not inserted by convention at this stage. It is required so that the interior product return df rather than −df. Hamilton’s equations are the integral curves of . The Poisson bracket is the pairing of the form on the two vector fields,

(15.151)

Antisymmetry of the bracket is antisymmetry of ω. Jacobi’s identity is the closure dω = 0, read on three Hamiltonian vector fields. Liouville’s vanishing divergence is the same closure, read as the invariance of the volume built from ω.

Figure 15.24: From a function to its vector field. The differential df is a one-form. Nondegeneracy supplies exactly one vector such that . On the phase plane that vector is the gradient of f rotated by a right angle, so that it circulates along the levels of f instead of climbing them.

Principle 15.5. Darboux charts. About every point of a symplectic manifold there are coordinates in which These are the canonical charts of the previous section. Darboux’s theorem says they are not a special gift of cotangent bundles. Any symplectic manifold looks like in a neighborhood of each point. The global topology may differ. The local algebra of brackets does not.

A symplectic form can be brought to the standard pairing in a whole neighborhood. Distinctions between symplectic manifolds are global, or else they sit in the Hamiltonian, not in ω.

Principle 15.6. The group at a point. The linear maps that preserve ω at a single tangent space are exactly the matrices M with . That is the symplectic group already defined. A canonical transformation is a diffeomorphism Φ such that . Its derivative at each point lies in that group. Preservation of the bracket, preservation of ω, and preservation of the Hamiltonian form of the equations are one condition, read on functions, on forms, and on vector fields.

Principle 15.7. The volume. The n-fold wedge

(15.152)

is a nowhere-vanishing 2n-form, the Liouville volume. In a canonical chart it is . A map that pulls ω back to itself pulls Ω back to itself, so every canonical transformation preserves volume. The converse remains false, as a previous section showed: volume is one scalar condition, and ω imposes n(2n − 1) independent conditions at each point. For n = 1 the two agree, which is why area on the phase plane was a faithful picture and must not be trusted as the whole story in more degrees of freedom.

Figure 15.25: What closure buys. Because dω = 0, the flow of drags ω onto itself. Areas of parallelograms spanned by pairs of vectors are unchanged, not merely the total volume of a cloud. The cloud may shear. The pairing inside the cloud does not.

The symplectic form does not choose the Hamiltonian. Many Hamiltonians share one ω; the form is the stage, H is the play. The form does not by itself give a metric, a notion of length, or a gradient. Rotating df to obtain uses ω, not an inner product, which is why circulates and does not descend. And Darboux’s theorem is local. A sphere of dimension 2 carries a symplectic form, the area form, and is not a cotangent bundle. Global models remain a later question.

Further Reading

Ian D. Lawrie (2112), A Unified Grand Tour of Theoretical Physics, CRC Press. This is a fairly advanced overview of modern theoretical physics.

Joseph L. McCauley, (1997), Classical Mechanics, Cambridge University Press, reprinted with correction in 1998. This is a really good introduction to advanced mechanics.

Jorge V. José and Eugene J. Saletan, (1998), Classical Dynamics: A Contemporary Approach, Cambridge University Press. A fairly complete study of geometrical mechanics.