Lecture 1 left two jobs. The Liouville–Arnold theorem (Lecture 5) needs $n$ conserved quantities whose flows commute, but Theorem 1.9 speaks of vector fields, not of conserved quantities. And the action variables of Lecture 6 are loop integrals of $p\cdot\dd q$, which need an integrand that a curve can integrate without a metric. So the question is

The answers are the Poisson bracket (§§2.2 and 2.3) and differential forms (§§2.4 and 2.5).

The Jacobi identity, proved. Write $z^{1},\dots,z^{2n}$ for all the coordinates at once and $$\pb{u}{v}=\sum_{\alpha,\beta}\partial_\alpha u\;J^{\alpha\beta}\;\partial_\beta v, \qquad J^{\alpha\beta}=\pb{z^\alpha}{z^\beta},$$(2.3) so that $J$ is a constant and antisymmetric matrix — that is all of (2.1) the argument uses. Expand the cyclic sum. Every term carries a second derivative of exactly one of $f,g,h$, so the terms fall into three classes, and it is enough to kill one: the others follow by relabelling $f,g,h$ cyclically. Take the class with a second derivative of $h$. It comes only from the first two summands, since $\pb{h}{\pb{f}{g}}$ differentiates $h$ just once, and it is $$\underbrace{\partial_\alpha f\,J^{\alpha\beta}\,\partial_\gamma g\, J^{\gamma\delta}\,\partial_\beta\partial_\delta h}_{\text{from }\pb{f}{\pb{g}{h}}} \;+\; \underbrace{\partial_\alpha g\,J^{\alpha\beta}\,\partial_\beta\partial_\gamma h\, J^{\gamma\delta}\,\partial_\delta f}_{\text{from }\pb{g}{\pb{h}{f}}} ,$$ summed over repeated indices. Relabel the dummy indices of the second term by $\alpha\mapsto\gamma$, $\beta\mapsto\delta$, $\gamma\mapsto\beta$, $\delta\mapsto\alpha$: it becomes $\partial_\alpha f\,\partial_\gamma g\,\partial_\beta\partial_\delta h\, J^{\gamma\delta}J^{\beta\alpha}$. The two terms now differ only in carrying $J^{\alpha\beta}$ where the other carries $J^{\beta\alpha}$, so their sum has the factor $J^{\alpha\beta}+J^{\beta\alpha}$, which vanishes. The class is zero, and so are the other two.

The bracket is what turns §1.3’s geometry back into calculation. For any observable $f$, the chain rule along a trajectory together with (1.6) gives $$\frac{\dd f}{\dd t} =\sum_i\left(\frac{\partial f}{\partial q_i}\dot q_i +\frac{\partial f}{\partial p_i}\dot p_i\right) =\pb{f}{H}.$$(2.4)

So $H$ generates time evolution. $\dd f/\dd t=\pb{f}{H}$, with the bracket in that order, is the convention of this course, and every sign that follows comes from it.

The condition $\pb{f}{H}=0$ is antisymmetric in its two arguments, so it vanishes for the pair $(f,H)$ exactly when it vanishes for $(H,f)$. Reading (2.4) both ways:

That is Noether’s theorem, at this level of generality, and it is a one-line consequence of antisymmetry. A conserved quantity is a symmetry; they are not two facts related by a theorem but one fact read in two directions. Angular momentum is conserved because $H$ is rotation-invariant, and — the same sentence — rotations are a symmetry because angular momentum is conserved.

Lecture 1 attached a vector field to $H$. The same construction works for any function.

Taking $f=H$ gives $\hvf{H}h=\pb{h}{H}=\dd h/\dd t$ by (2.4), so $\hvf{H}$ is the field (1.7) of Lecture 1. The map $f\mapsto\hvf{f}$ takes functions, with their Poisson bracket, to vector fields, with their Lie bracket. The question is whether it respects the two brackets.

Split the oscillator’s Hamiltonian, at unit frequency, as $H=\tfrac12(q^{2}+p^{2})=f+g$ with $f=q^{2}/2$ and $g=p^{2}/2$. Their fields, the Lie bracket of the fields, and the field of their Poisson bracket can each be computed in a line: $$\begin{aligned} \hvf{f}&=-q\,\partial_p, &\qquad \hvf{g}&=p\,\partial_q, &\qquad \comm{\hvf{f}}{\hvf{g}}&=-q\,\partial_q+p\,\partial_p,\\[2pt] \pb{f}{g}&=qp, &\qquad \hvf{qp}&=q\,\partial_q-p\,\partial_p, &\qquad \hvf{\pb{f}{g}}&=-\comm{\hvf{f}}{\hvf{g}}. \end{aligned}$$(2.6) So here the map respects the brackets up to a sign, which §2.7 explains. The sum of the two halves is the oscillator, and $\hvf{H}=\hvf{f}+\hvf{g}=p\,\partial_q-q\,\partial_p$ is the rotation of the phase plane; its orbits other than the equilibrium are circles, the positive-energy level curves of §1.4 at unit frequency. The engine says that what happened here happens for every pair of functions.

The proof is a computation in coordinates. A second proof, three lines long and from the Jacobi identity, shows why it is true; it is a reading.

The second proof locates the content: the engine is the Jacobi identity, wearing different clothes. The map $f\mapsto\hvf{f}$ is an anti-homomorphism of Lie algebras, and it is one because the bracket satisfies Jacobi — which is also, as a reading of Lecture 4 shows, the same statement as $\dd\symp=0$.

Only two properties of $J$ were used, that it is constant and antisymmetric, so the Jacobi identity has nothing to do with the particular pairing of each $q$ with its own $p$. Lecture 3 meets the same fact in geometric clothes: antisymmetry becomes $\symp$ being a $2$-form, and constant coefficients make $\dd\symp=0$ automatically. Closedness without constancy is enough, and it is what corresponds to Jacobi in general.

What the corollary buys is large. It converts an algebraic hypothesis that can be checked — $n$ functions whose brackets vanish — into a geometric conclusion: $n$ commuting vector fields, each tangent to the common level sets of the functions, since $\hvf{F_j}F_i=\pb{F_i}{F_j}=0$. On a compact regular component of a level set the fields are complete (Proposition 1.7, applied to the component), so their flows commute and together give an action of the abelian group $\R^{n}$ on it. That action is the whole machine of the Liouville–Arnold theorem in Lecture 5, and Theorem 2.5 is what starts it.

In Lecture 6 we shall meet the integral $$I \;=\; \frac{1}{2\pi}\oint_{\gamma} p\cdot\dd q \;=\; \frac{1}{2\pi}\oint_{\gamma}\ \sum_{i=1}^{n} p_i\,\dd q_i ,$$(2.8) taken around a closed loop $\gamma$ on an invariant torus, and everything in Lectures 5 and 6 depends on it being a well-defined number. Three separate things have to be true, and not one of them is obvious:

1. The integrand must be an object that a curve can integrate, with no extra structure supplied. In particular no length, no angle, no inner product: phase space has no metric, and any construction that secretly uses one is not available to us.

2. The answer must not change when the curve is reparametrised. A loop traversed quickly and the same loop traversed slowly must give the same number.

3. The answer must not change when $\gamma$ is deformed. Otherwise (2.8) labels a curve rather than a torus, and there is no “action variable” at all.

This lecture settles the first by getting the type of the integrand right: $p\cdot\dd q$ is a $1$-form. Lecture 3 settles the second, since line integrals of $1$-forms do not see the parametrisation, and proves what the third needs: the integral of a closed $1$-form round a loop does not change when the loop is deformed. That $p\cdot\dd q$ is closed on the tori where (2.8) is taken — compact connected components of common level sets of $n$ conserved quantities in involution, on which their differentials are independent — is proved in Lecture 5, and Lecture 6 then defines the actions.

Throughout this lecture $\phase$ is a smooth manifold of dimension $m$, and $x^1,\dots,x^m$ is a chart on it. In the application $\phase$ is phase space and $m=2n$, with $n$ the number of degrees of freedom — $n$ is reserved for that throughout the course, so it is not available here.

Recall from §1.2 that a tangent vector at $p\in\phase$ is the velocity of a curve through $p$, equivalently a derivation on functions near $p$, and that these form an $m$-dimensional vector space $T_p\phase$. The chart gives $T_p\phase$ the basis $\partial/\partial x^1,\dots,\partial/\partial x^m$.

The example to hold on to is the differential of a function.

Applying (2.9) to the coordinate functions $x^i$ themselves gives $\dd x^i(\partial/\partial x^j)=\partial x^i/\partial x^j=\delta^i_j$: the $1$-forms $\dd x^1,\dots,\dd x^m$ are exactly the basis of $T^{*}_p\phase$ dual to the coordinate basis of $T_p\phase$. Hence every $1$-form is $$\alpha \;=\; \sum_{i} a_i(x)\,\dd x^i , \qquad\text{and in particular}\qquad \dd f \;=\; \sum_i \frac{\partial f}{\partial x^i}\,\dd x^i ,$$(2.10) the second identity being (2.9) evaluated on each basis vector. So the notation $\dd f$, which until now was a formal symbol in the chain rule, is a genuine object: a machine that eats a tangent vector and returns a number.

A tangent vector is drawn as an arrow. A covector is not an arrow; it is a stack. Draw the level sets $f=c$, $f=c+1$, $f=c+2$, … near $p$: for a nearly-constant $f$ they are far apart, for a rapidly varying $f$ they are close together (Figure 2.1). Then (2.9) says

Figure 2.2 shows why no metric is needed. Nothing in the counting used how long $v$ is or what angle it makes with anything; a covector is exactly the structure that can be evaluated on a vector with no metric present. Addition of covectors is superposition of stacks (Figure 2.3): $\alpha+\beta$ is denser than either.

A $1$-form measures a curve’s worth of information: one vector in, one number out. To integrate over a surface we need a machine taking two vectors, and it should return the area of the parallelogram they span — because that is what a surface integral adds up.

Start in the plane, where the answer is already known from linear algebra: the parallelogram spanned by $v=(v^1,v^2)$ and $w=(w^1,w^2)$ has signed area $$A(v,w)\;=\;\det\begin{pmatrix} v^1 & w^1\\ v^2 & w^2\end{pmatrix} \;=\; v^1w^2-v^2w^1 .$$(2.11) Three properties of $A$ are worth naming before we generalise, because they are forced by what “area” means and not chosen for convenience.

• Bilinearity. Doubling one edge doubles the area; gluing two parallelograms along an edge adds their areas.

• $A(v,v)=0$. A degenerate parallelogram has no area.

• $A(w,v)=-A(v,w)$. The area is signed: it records not only how much area there is but which way round the two edges run.

The third property is the one students find arbitrary, so note that it is not an extra assumption: it follows from the first two. Expanding $0=A(v+w,v+w)=A(v,v)+A(v,w)+A(w,v)+A(w,w)$ gives $A(v,w)=-A(w,v)$ at once. Antisymmetry is what you get for free once you ask that a degenerate parallelogram have zero area.

Two $1$-forms produce a $2$-form in exactly one natural way.

Antisymmetry in $(v,w)$ is manifest, so $\alpha\wedge\beta$ is a $2$-form; and swapping the two forms transposes the determinant, so $$\alpha\wedge\beta=-\beta\wedge\alpha, \qquad\text{and in particular}\qquad \alpha\wedge\alpha=0 .$$(2.13) This is the same minus sign as Figure 2.4, now in the algebra. It is worth saying out loud that (2.13) is a statement about orientation, not a convention about symbols: a plane with a circulation arrow drawn on it is a different object from the same plane with the arrow reversed, and $\wedge$ is the product that knows the difference.

In $m$ dimensions, a $2$-form has components $\beta_{ij}=\beta(\partial_i,\partial_j)$ with $\beta_{ij}=-\beta_{ji}$, so $$\beta \;=\; \sum_{i<j}\beta_{ij}\;\dd x^i\wedge \dd x^j ,$$(2.14) a sum over $\binom{m}{2}$ independent components. In the plane there is one, $\dd x\wedge\dd y$; in $\R^{3}$ there are three; in $\R^{4}$, six — and it is the $\R^{4}$ count that Lecture 3 will need when it writes $\symp=\dd q_1\wedge\dd p_1+\dd q_2\wedge\dd p_2$.

The pattern of (2.12) continues. A $k$-form is a multilinear map on $k$ tangent vectors that changes sign whenever two arguments are swapped, and the wedge of $k$ $1$-forms is the determinant of the matrix of pairings: $$(\alpha^{1}\wedge\cdots\wedge\alpha^{k})(v_1,\dots,v_k) \;=\;\det\!\big[\alpha^{i}(v_j)\big]_{i,j=1}^{k} .$$(2.15) One more rule is needed, because (2.15) covers only wedges of $1$-forms: the wedge of forms of any degree, which the exterior derivative of Lecture 3 uses.

• A test for commuting flows that can be computed. Functions whose brackets vanish have commuting fields (Corollary 2.6), and commuting flows when the fields are complete. That is the hypothesis of Lecture 5, in a form that a calculation can settle.

• The type of $p\cdot\dd q$. It is a $1$-form (Example 2.9), so a curve can integrate it with no metric. Lecture 3 differentiates and integrates forms, and proves that the loop integral of a closed $1$-form depends on the loop only through its class; Lecture 5 shows that $p\cdot\dd q$ is closed on the tori where it will be integrated. Lecture 3 and Lecture 4 then show that the two halves of this lecture are one structure, the symplectic form, a $2$-form.

• Status. The Jacobi identity in Proposition 2.2 is proved in a reading; the rest of that proposition, the engine and its corollary are proved in the lectured core, except that the corollary’s last clause, commuting flows, rests on Theorem 1.9, which is cited.

• The sign. (2.7) carries a minus sign because of the convention $\dd f/\dd t=\pb{f}{H}$ of (2.4), together with (2.5). Sources that write $\hvf{f}h=\pb{f}{h}$, or put the bracket of the evolution law the other way round, print a plus, and they make no different claim about mechanics. Corollary 2.6, the consequence that the proof of the Liouville–Arnold theorem uses, does not see the sign.

• Coordinates. The bracket (2.1) is defined here on $T^{*}\R^{n}$, in the coordinates $(q,p)$. That the same bracket makes sense on any phase space, in any canonical chart, is Lecture 4’s.