Lecture 2 gave $p\cdot\dd q$ a type, a $1$-form, and left two requirements on the loop integral (2.8) open: it must not change when the loop is reparametrised, and it must not change when the loop is deformed. So the question of this lecture is

The answer to the first is closedness, with Stokes’ theorem (Corollary 3.13). The answer to the second is the symplectic form $\symp$, of which $p\cdot\dd q$ is a primitive up to sign (§3.6).

We have $\dd$ on functions. Extending it to forms of every degree is what turns this algebra into a calculus.

The formula is written in coordinates, so it is fair to ask whether a different chart would give a different answer. It would not: $\dd$ is characterised by (3.2), by agreeing with Definition 2.8 in degree zero, and by $\dd\circ\dd=0$, and those three conditions mention no coordinates. We take the characterisation on trust (cited; da Silva, §6) and use the formula.

In the plane, for $\alpha=P\,\dd x+Q\,\dd y$, (3.1) gives $$\dd\alpha=\Big(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\Big)\, \dd x\wedge \dd y ,$$(3.3) using $\dd x\wedge\dd x=0$ and $\dd y\wedge\dd x=-\dd x\wedge\dd y$. Evaluated on a small parallelogram, $\dd\alpha$ is the circulation of $\alpha$ around its boundary (Figure 3.1). That reading is Stokes’ theorem in miniature, and §3.4 does no more than add such squares up.

Theorem 3.2 says exact $\Rightarrow$ closed. The converse is the interesting direction, and it is local.

Everything this course does with forms lives in the gap between Theorem 3.2 and Theorem 3.4: closed and exact differ, and they differ only because of the shape of the region (Figure 3.2). §3.3 is the smallest example, and Lecture 6 is the payoff.

In $\R^{3}$ the forms of each degree have three, three and one independent components — and so do the classical vector operators. Writing $\dd$ in each degree reproduces them:

Read in this table, Theorem 3.2 is two familiar identities at once: $\operatorname{curl}\operatorname{grad}=0$ and $\operatorname{div}\operatorname{curl}=0$. They were never two facts. The dictionary is special to three dimensions — it relies on $3$-forms and $1$-forms both having three components — which is the reason the classical operators do not generalise and $\dd$ does. Phase space is $2n$-dimensional, so from here on only $\dd$ is available — which is why this comparison is worth making once and then leaving behind.

In polar coordinates $\alpha=\dd\theta$, which makes it look exact. It is not, and the reason is that $\theta$ is not a function on the annulus: continuing it once around the hole returns it increased by $2\pi$. The name $\dd\theta$ is a name for the $1$-form (3.4), not an assertion that something has been differentiated. To see this is not a quibble, integrate.

This is the smallest interesting fact in the lecture, and the one to remember. It is tempting to read Proposition 3.6 as bookkeeping. It is not:

The two halves do different jobs and neither can be dropped. Were $p\cdot\dd q$ exact on the torus, (2.8) would give $I=0$ for every cycle on it, by Proposition 3.10. The nonzero actions of the oscillator, the pendulum and the Kepler problem exist for the reason Figure 3.3 exhibits — a cycle that does not bound, and a closed form with no primitive along it.

Forms have one property vectors do not: they can be transported backwards along any smooth map, with no assumption that the map be invertible.

In practice pullback is substitution: if $F$ is given by $x^i=x^i(u)$, then $F^{*}(\dd x^i)=\sum_a(\partial x^i/\partial u^a)\,\dd u^a$, and one simply substitutes. Three rules follow, used later and the reason the formalism is worth its cost: $$F^{*}(\dd f)=\dd(f\circ F), \qquad F^{*}(\alpha\wedge\beta)=F^{*}\alpha\wedge F^{*}\beta, \qquad F^{*}(\dd\alpha)=\dd (F^{*}\alpha).$$(3.6) The first is the chain rule, and the second is the definition applied pointwise. The third follows from them for a $1$-form, the case the deformation argument below uses: with $\alpha=\sum_i a_i\,\dd x^i$ and $F^{*}\alpha=\sum_i (a_i\circ F)\,\dd(x^i\circ F)$, the graded Leibniz rule (3.2) and $\dd\dd(x^i\circ F)=0$ give $\dd(F^{*}\alpha)=\sum_i\dd(a_i\circ F)\wedge\dd(x^i\circ F) =F^{*}\big(\sum_i\dd a_i\wedge\dd x^i\big)=F^{*}(\dd\alpha)$. The last rule — $\dd$ commutes with pullback — is what makes “closed” and “exact” coordinate-independent notions, and Lecture 4 will use the middle one to recognise canonical transformations as the maps satisfying $F^{*}\symp=\symp$.

Now the type-checking of §2.4 pays off.

So the answer depends on the curve and its direction, not on the schedule. That was requirement 2 of §2.4, and Definition 3.8 met it by construction: a $1$-form is precisely the kind of object whose integral over a curve needs no further data.

Proposition 3.10 is what Proposition 3.6 used, and it is also the statement that an exact form carries no information about loops. Everything interesting therefore lies with forms that are closed and not exact.

We do not prove it. What we do instead is say exactly why it is true, because the mechanism is short and is the picture the rest of Part I uses.

This is requirement 3 of §2.4, discharged for closed forms: the integral of a closed $1$-form depends on the loop only through its class. That $p\cdot\dd q$ is closed on the invariant tori, in $\R^{2n}$ or a cotangent bundle, is Lecture 5’s. An action variable is attached not to a curve but to a class of curves, and Lecture 5 will produce exactly $n$ such classes on an $n$-dimensional invariant torus — the basis of cycles $\gamma_1,\dots,\gamma_n$ that Figure 3.6 shows for $n=2$, and Lecture 6 will integrate $p\cdot\dd q$ over each of them in turn.

Take the oscillator at unit frequency, $H=\tfrac12(p^{2}+q^{2})$, and its orbit $H=E>0$: the circle of radius $\sqrt{2E}$, traversed clockwise (§1.4), $q=\sqrt{2E}\cos t$, $p=-\sqrt{2E}\sin t$ for $t\in[0,2\pi]$. Then $$\oint_{\gamma}p\,\dd q=\int_{0}^{2\pi}\big(-\sqrt{2E}\sin t\big) \big(-\sqrt{2E}\sin t\big)\,\dd t=2\pi E .$$(3.9) Stokes gives the same number without a parametrisation. The clockwise orbit bounds the disc $D$ it encloses, but with the orientation opposite to the one $\dd q\wedge\dd p$ gives $D$; and $\dd(p\,\dd q)=\dd p\wedge\dd q=-\dd q\wedge\dd p$. The two signs cancel: $$\oint_{\gamma}p\,\dd q=-\int_{D}\dd(p\,\dd q)=\int_{D}\dd q\wedge\dd p =\text{area of }D=2\pi E .$$ The loop integral is the area enclosed, and it is positive. Lecture 6 calls $\frac{1}{2\pi}$ of it the oscillator’s action, $I=E$.

Lecture 2 built $2$-forms, and §3.2 showed how to tell a closed one. The phase space of mechanics carries one closed $2$-form with a special property, and the rest of Part I is built on it. One piece of notation first: a $2$-form eats two vectors, and feeding it one leaves a $1$-form.

Condition 2 says that the linear map $v\mapsto\iota_v\symp$ from $T_p\phase$ to $T_p^{*}\phase$ has zero kernel. The two spaces have the same dimension, so $$v\;\longmapsto\;\iota_v\symp \quad\text{is an isomorphism } T_p\phase\;\xrightarrow{\;\sim\;}\;T_p^{*}\phase .$$(3.10) A symplectic form converts vectors into covectors and back, with no choices made — the job an inner product does in Euclidean space, done by an antisymmetric object.

The wedge order $\dd q\wedge\dd p$ is not a matter of taste. Ask which $X$ satisfies $\iota_X\symp=\dd H$ on $\R^{2}$. By Definition 3.14 and (2.12), $$\iota_X(\dd q\wedge\dd p)=\dd q(X)\,\dd p-\dd p(X)\,\dd q = X^{q}\,\dd p-X^{p}\,\dd q ,$$ and matching with $\dd H=H_q\,\dd q+H_p\,\dd p$ gives $$X^{q}=\frac{\partial H}{\partial p},\qquad X^{p}=-\frac{\partial H}{\partial q}:$$(3.13) Hamilton’s equations, both signs right; for the pendulum $H=\tfrac12p^{2}-\cos q$ this is $\dot q=p$, $\dot p=-\sin q$. Had we written $\dd p\wedge\dd q$, the same computation would return $X^{q}=-H_p$ and $X^{p}=+H_q$ — both signs wrong. That is the whole reason for the order.

• The action variables can be defined. A closed $1$-form integrates to the same number over any two loops that deform into one another (Corollary 3.13). On the invariant tori of Lecture 5, in $\R^{2n}$ or a cotangent bundle, $p\cdot\dd q$ is closed, and Lecture 6 integrates it over each torus’s $n$ independent cycles (Figure 3.6).

• $p\cdot\dd q$ and $\symp$ are one object. On $T^{*}\R^{n}$, by (3.1) and (2.13), $$\dd\Big(\sum_i p_i\,\dd q_i\Big)=\sum_i\dd p_i\wedge\dd q_i=-\symp ,$$ so the standard symplectic form is exact, and closed by Theorem 3.2. The minus sign comes with the wedge order $\dd q\wedge\dd p$ that the sign test fixed. Lecture 4 shows the same on every cotangent bundle, where $p\cdot\dd q$ needs no coordinates.

• A symplectic form turns functions into flows. By (3.10), $\iota_X\symp=\dd H$ has exactly one solution $X$, and on $\R^{2}$ it is Hamilton’s vector field (3.13). Lecture 4 makes this the definition of $\hvf{H}$ on any symplectic manifold, and shows that it agrees with Lecture 2’s.

• Status. Three results are cited: that $\dd$ does not depend on the chart, the Poincaré lemma (Theorem 3.4), and Stokes’ theorem (Theorem 3.11). Deformation invariance (Corollary 3.13) is proved from Stokes, so it is proved modulo a cited theorem, and so is everything later that rests on it. Reparametrisation invariance (Proposition 3.9) is proved in a reading.

• A deformation carries orientations. Corollary 3.13 needs a deformation, not merely two loops that together bound a region. The circle $r=2$ traversed anticlockwise and the circle $r=1$ traversed clockwise bound the annulus between them as point sets, yet the angle form (3.4) integrates to $2\pi$ over the first and to $-2\pi$ over the second.