Lecture 2 defined the Hamiltonian vector field from the Poisson bracket, and Lecture 3 defined the symplectic form. The Liouville–Arnold theorem of Lecture 5 needs both at once: it is about the flows of $n$ functions in involution, and about the shape of the level set they share. So the question is

The structure is $\symp$ itself (§4.2). The maps that preserve it are the canonical transformations (§4.3), and the submanifolds it singles out are the Lagrangian ones (§4.4). The worked example is the two-dimensional oscillator, whose level sets are Lagrangian tori (Example 4.15).

Lecture 2 defined $\hvf{f}$ from the bracket, by $\hvf{f}h=\pb{h}{f}$ (Definition 2.4). On a symplectic manifold there is a second route, which uses no bracket at all.

On Lecture 2’s worked example, $f=q^{2}/2$ and $g=p^{2}/2$ with $\hvf{f}=-q\,\partial_p$ and $\hvf{g}=p\,\partial_q$ (2.6), $$\symp(\hvf{f},\hvf{g}) =\dd q(\hvf{f})\,\dd p(\hvf{g})-\dd q(\hvf{g})\,\dd p(\hvf{f}) =0\cdot0-p\cdot(-q)=qp=\pb{f}{g}.$$

Turn (4.2) round and make it a definition: on any symplectic manifold, put $$\pb{f}{g}\;:=\;\symp(\hvf{f},\hvf{g})=\dd f(\hvf{g}) .$$(4.3) It is bilinear and antisymmetric, and satisfies Leibniz because $\dd(fg)=f\,\dd g+g\,\dd f$. What is left is Jacobi, and this is where closedness enters.

The Jacobi sum turns out to be $\dd\symp(\hvf{f},\hvf{g},\hvf{h})$, a computation done in the reading below. So the bracket (4.3) satisfies the Jacobi identity exactly when $\dd\symp=0$: that is what closedness in Definition 3.15 is for. With Jacobi in hand, the second proof of the engine in Lecture 2 runs unchanged, so Theorem 2.5 and Corollary 2.6 hold on every symplectic manifold.

Jacobi is $\dd\symp=0$, proved.

The proof is Lecture 2’s computation of the Jacobi identity with non-constant coefficients: the terms in which a derivative lands on a coefficient no longer vanish, and they assemble into exactly $\dd\symp$.

Lecture 2 proved the engine $\hvf{\pb{f}{g}}=-\comm{\hvf{f}}{\hvf{g}}$ from the Jacobi identity alone (Theorem 2.5, second proof, a reading). So the engine runs on any closed non-degenerate $2$-form, and on nothing less: that is the sense in which Lecture 2’s bracket and its forms are one structure.

In canonical coordinates, with $F$ written as $(Q,P)=F(q,p)$, the condition is $$\sum_i\dd Q_i\wedge\dd P_i\;=\;\sum_i\dd q_i\wedge\dd p_i ,$$(4.5) where on the left $Q$ and $P$ are regarded as functions of $(q,p)$ and substituted — pullback is substitution (§3.4). It is a condition you already know under another name, since the matrix of the pulled-back form is $D^{\mathsf T}\Omega D$, with $D$ the Jacobian of $F$.

For one degree of freedom $\dd Q\wedge\dd P=(\det D)\,\dd q\wedge\dd p$, so a canonical transformation of the plane is exactly an area- and orientation-preserving map. To check the shear $(q,p)\mapsto(q+p^{2},p)$, pull back: $\dd(q+p^{2})\wedge\dd p=\dd q\wedge\dd p+2p\,\dd p\wedge\dd p=\dd q\wedge \dd p$. The stretch $(q,p)\mapsto(2q,p)$ fails, with $\pb{2q}{p}=2$. For $n\ge2$ the condition is much stronger than preserving volume (a reading at the end of this lecture).

On a general symplectic manifold the trajectory may leave any one chart, but $\varphi_{t+s}=\varphi_s\circ\varphi_t$ gives $$\frac{\dd}{\dd t}\,\varphi_t^{*}\symp =\varphi_t^{*}\Big(\frac{\dd}{\dd s}\Big|_{s=0}\varphi_s^{*}\symp\Big),$$ and the bracket is a rate of change at a single point, which may be computed in a Darboux chart there (Theorem 4.18): it is the computation above at $t=0$, where $D=\Id$, and it vanishes.

With canonical brackets, Leibniz and the chain rule give $\dot Q_i=\pb{Q_i}{H}=\partial H/\partial P_i$ and $\dot P_i=-\partial H/\partial Q_i$: Hamilton’s equations keep their form.

A transformation given as a formula is easy to check. The problem in practice runs the other way: to produce canonical transformations adapted to a given $H$. Generating functions do this, and the form language says in one line why they work.

That is the mechanism in full: a generating function is a primitive for a combination of canonical $1$-forms, and $\dd\circ\dd=0$ turns “exact” into “canonical”. For example $S=\sum_iq_iP_i+f(q)$ gives $Q=q$ and $P=p-\nabla f$: shifting every momentum by a gradient is canonical. The function $\sum_i q_iP_i$ alone gives the identity.

The other classical types work identically. A type-1 function $S(q,Q)$ with $p=\partial S/\partial q$ and $P=-\partial S/\partial Q$ says $p\cdot\dd q-P\cdot\dd Q=\dd S$; for instance $S=qQ$ gives $(Q,P)=(p,-q)$, which exchanges position and momentum up to a sign. Type 1 cannot generate the identity, whose $q$ and $Q$ are not independent, which is one reason type 2 is the workhorse and the type Lecture 6 uses.

Hamilton–Jacobi looks for a type-2 generating function adapted to $H$: an $S(q,P)$ with $$H\Big(q,\frac{\partial S}{\partial q}\Big)=K(P)$$(4.7) for some function $K$ of the new momenta alone. The new equations are then $\dot P_i=-\partial K/\partial Q_i=0$ and $\dot Q_i=\partial K/\partial P_i$, a constant: conserved momenta, and positions advancing at constant speed. Lecture 6 does this with $P=I$ the actions and $Q=\varphi$ the angles. With $P$ held fixed, (4.7) also says that the $n$-dimensional set $\{(q,\partial S/\partial q)\}$ lies inside a level set of $H$ — §4.4 returns to this.

Let $\dim V=2n$ with $\symp$ non-degenerate. The $\symp$-complement of a subspace $U$ is $U^{\symp}=\{v:\symp(v,u)=0\text{ for all }u\in U\}$; by (3.10) it is the preimage of the covectors vanishing on $U$, so $$\dim U^{\symp}=2n-\dim U .$$(4.8) Unlike an orthogonal complement, $U^{\symp}$ can contain $U$, since every vector is $\symp$-orthogonal to itself. Call $U$ isotropic if $\symp$ vanishes on it, i.e. $U\subseteq U^{\symp}$; then (4.8) gives $\dim U\le 2n-\dim U$: an isotropic subspace has dimension at most $n$.

This bound is why Lecture 5 asks for exactly $n$ integrals.

In $\R^{4}$, $\symp(v,w)=A_1+A_2$, the signed areas of the two shadows of the parallelogram on $v,w$. A plane is Lagrangian when the two shadow areas of a basis parallelogram cancel (Figure 4.2).

The most important examples are graphs, and they bring back Lecture 3.

$S$ is called a generating function of the Lagrangian submanifold, and the name is not a coincidence: by (4.7), a solution of Hamilton–Jacobi is a family of Lagrangian graphs, one for each value of $P$, each lying in a level set of $H$. Lecture 6’s $S(q,I)$ is exactly that, with the invariant tori as the family.

By the Poincaré lemma a closed $\sigma$ is locally some $\dd S$; globally it need not be. On the pendulum’s cylinder, with angle $q$, a rotation orbit at energy $E>1$ is the graph $p=\sqrt{2(E+\cos q)}$ of the $1$-form $\sigma=\sqrt{2(E+\cos q)}\,\dd q$ — closed, as every $1$-form on a circle is, and Lagrangian. It is not $\dd S$ for any function $S$ on the circle, for the reason of Proposition 3.6: $\oint\sigma>0$. Lecture 6 calls $\frac{1}{2\pi}\oint\sigma$ the rotating pendulum’s action. A nonzero action is precisely the obstruction to a global generating function, and Lecture 6 lives in that gap.

Example 4.15 is Lecture 5’s first proof step, done by hand; the general argument is Proposition 4.11’s, with equality. Like the circle of Figure 4.3, $T$ is not a graph over $(q_1,q_2)$.

Let $\config$ be a configuration space, a manifold of dimension $n$ — the circle for the pendulum, the torus $T^{2}$ for the double pendulum. Its cotangent bundle is the set of pairs $$T^{*}\config=\{(q,p):\ q\in\config,\ p\in T^{*}_{q}\config\}, \qquad \pi(q,p)=q ,$$ a point of configuration space and a covector there. A chart $q_1,\dots,q_n$ on $\config$ writes the covector as $p=\sum_ip_i\,\dd q_i$ (Definition 2.7), and $(q,p)$ is then a chart on the $2n$-dimensional $T^{*}\config$. That momenta are covectors shows in a change of chart $q'=\phi(q)$: writing the same covector in both bases, $\sum_j p'_j\,\dd q'_j=\sum_i p_i\,\dd q_i$, forces $$p_i=\sum_j p'_j\,\frac{\partial q'_j}{\partial q_i} :$$(4.10) momenta transform like the components of a gradient, not like those of a velocity. For polar coordinates on the plane, (4.10) gives $p_r=(x\,p_x+y\,p_y)/r$ and $p_\theta=x\,p_y-y\,p_x$ — the momentum conjugate to the angle is the angular momentum, as it should be.

In a chart, $\dd\pi\,v=\sum_iv^{q_i}\partial_{q_i}$ drops the momentum components of $v$, so (4.11) is $\big(\sum_ip_i\,\dd q_i\big)(v)$. The form of Example 2.9, and of Lecture 6’s action integral, is therefore the same $1$-form in every cotangent chart, because (4.11) mentions no chart.

So the pendulum’s cylinder, the double pendulum’s $T^{*}T^{2}$ and the rigid body’s $T^{*}SO(3)$ are symplectic, with no choice made.

The minus sign in $\symp=-\dd(p\cdot\dd q)$ comes with the wedge order $\dd q\wedge\dd p$, which (3.13) showed is the one giving Hamilton’s equations. It is compensated where it matters. Orbits of the oscillator run clockwise in the $(q,p)$-plane, a clockwise orbit bounds its disc with the opposite orientation, and the two signs cancel: $\oint p\,\dd q$ along the motion is $+\int_D\symp$, the enclosed area, which is positive (§3.4 computed it). So the action integral of Lecture 6, taken along the motion, comes out positive.

A diffeomorphism $\phi$ of configuration space moves momenta by the transpose of its inverse Jacobian — that is (4.10) read as a map — and so lifts to a map $\hat\phi(q,p)=(\phi(q),p')$ of $T^{*}\config$, the cotangent lift. Because $p\cdot\dd q$ is defined without a chart, $\hat\phi^{*}(p\cdot\dd q)=p\cdot\dd q$, and hence $\hat\phi^{*}\symp=\symp$ by (3.6). Every change of configuration coordinates, with the momenta carried along as covectors, is a canonical transformation. This is why passing to polar coordinates, with $p_\theta$ the angular momentum, never needs checking. It is also the type-2 generating function $S(q,p')=\sum_jp'_j\,\phi_j(q)$ of Proposition 4.9, as a one-line computation with (4.6) confirms.

• The count in the definition of integrability. At most $n$ independent functions Poisson-commute (Proposition 4.11), because their Hamiltonian vector fields span a subspace on which $\symp$ vanishes. So $n$ is the most one can ask for, and Lecture 5 asks for exactly that.

• The shape of the invariant tori. The oscillator’s level sets are Lagrangian tori (Example 4.15). Lecture 5 proves the same of every compact connected component of a common level set of $n$ functions in involution on which their differentials are independent.

• The tool for action–angle variables. A type-2 generating function $S(q,P)$ produces a canonical transformation (Proposition 4.9), and a solution of Hamilton–Jacobi is a family of Lagrangian graphs. Lecture 6 builds the action–angle variables from such an $S(q,I)$.

• A chart-free $p\cdot\dd q$. On any cotangent bundle the canonical $1$-form needs no coordinates (Definition 4.16), and $\symp=-\dd(p\cdot\dd q)$ (Theorem 4.17), so the action integral of Lecture 6 is the same in every chart.

• Status. Of the results the lectured core uses, three are proved only in readings: the Jacobi identity for the bracket of $\symp$, that symplectomorphisms are the maps with canonical brackets (Proposition 4.7), and that Hamiltonian flows are symplectic. So the engine and Corollary 2.6 on a general symplectic manifold rest on two reading proofs, this lecture’s and the second proof of the engine in Lecture 2. Darboux’s theorem is sketched, and its version adapted to commuting functions is cited, both in a reading. Propositions 4.2, 4.9, 4.11 and 4.14 and Theorem 4.17 are proved in the lectured core.

• Generating functions are local. Proposition 4.9 holds where $(q,P)$ are good coordinates. A closed $1$-form need not have a global primitive, and Lecture 6’s $S(q,I)$ changes branch at the turning points (Figure 4.3).

Such a chart is a Darboux chart, or canonical coordinates. So any statement about $\symp$ alone — or about the bracket and the fields $\hvf{f}$ built from it — that is proved in $q$’s and $p$’s holds near every point of every symplectic manifold.

Compare a Riemannian metric, which measures lengths and angles. A sphere of radius $R$ has curvature $1/R^{2}$ at every point, and curvature is a local invariant: no chart, however small, makes the sphere’s metric into the flat $\dd x^{2}+\dd y^{2}$. Darboux’s theorem says a symplectic form has nothing of the kind (Figure 4.6): every small patch of every $2n$-dimensional symplectic manifold is indistinguishable from every other, and the dimension is the only local invariant there is.

It is easy to hear “no local invariants” as “nothing interesting happens”, and that is the opposite of the point: everything interesting is global. It lives in loops that do not bound, in regions of finite size, in how charts are glued together — never at a point. The actions of Lecture 6 are integrals over loops, and the tori of Lecture 5 are compact; neither is visible in a Darboux chart.

At a single point, linear algebra already makes $\symp$ standard (the reading proves this). Call that constant form $\symp_0$ and join it to $\symp$ by the straight line $\symp_t=\symp_0+t(\symp-\symp_0)$, $0\le t\le1$. Near the point each $\symp_t$ is non-degenerate, and $\symp-\symp_0$ is closed, hence $\symp-\symp_0=\dd\sigma$ by the Poincaré lemma (Theorem 3.4). Now look for a flow $\psi_t$ that pulls every $\symp_t$ back to $\symp_0$. Its velocity $V_t$ is forced: $\iota_{V_t}\symp_t=-\sigma$, which non-degeneracy solves. That this $V_t$ does the job is one differentiation, which uses $\dd\symp_t=0$ and a formula for differentiating a form along a flow that we cite (reading). Then $\psi_1$ carries $\symp$ to $\symp_0$, and its coordinates are a Darboux chart.

Lemma 4.19 makes $\symp$ standard at one point by a linear change of coordinates. A metric can be pushed a little further — suitable coordinates make it Euclidean at a point to first order — but no further: at second order its curvature is in the way. A symplectic form can be made standard on a whole neighbourhood, and that is Theorem 4.18. The proof spends closedness in exactly one place.

In such a chart the $F_i$ are momenta and the $\psi_i$ positions, so by Proposition 4.2 $\hvf{F_j}=\partial/\partial\psi_j$: the commuting flows are translations at unit speed. That is the whole of the Liouville–Arnold theorem locally, and it costs nothing — as Babelon, Bernard and Talon remark (ch. 2, Remark 2), near any point where $\dd H\ne0$ Darboux gives canonical coordinates with $H=P_1$, so every Hamiltonian is “integrable” on a small enough patch.

So Lecture 5 is not trivial. Its question is whether the $\psi_i$ continue over a whole level set, and what they do when they are continued: they come back to themselves, which makes them angles and the level set a torus. Action-angle coordinates are an adapted Darboux chart on a whole neighbourhood of a torus, and the global part is what Lecture 5 and Lecture 6 add.

This is Liouville’s theorem on phase-space volume (an exercise), now a consequence of the stronger $\varphi_t^{*}\symp=\symp$. For $n=1$ the two coincide; for $n\ge2$ the symplectic statement is strictly stronger.

The proof, Stokes on the tube the loop sweeps out (Figure 4.7), is in the reading.

The same holds on any cotangent bundle, since the proof used only two facts that hold there globally: $\dd(p\cdot\dd q)=-\symp$ (Theorem 4.17) and $\iota_{\hvf{H}}\symp=\dd H$ (Definition 4.1).