Functions in involution have commuting Hamiltonian vector fields, whose flows commute where they are complete (Corollary 2.6), and at most $n$ independent functions can be in involution (Proposition 4.11). A system with $n$ of them has as many conserved quantities as it can have. So the question is

The answer, Theorem 5.7, concerns a connected component of a common level set on which the integrals are independent at every point. If that component is compact, the motion is a straight line on a torus; if it is not, but the flows on it are complete, it is a straight line on a cylinder. The worked example is the two-dimensional oscillator (Example 5.3), whose level sets with both energies positive are the tori of Example 4.15.

A first integral of $H$ is a function $F$ with $\pb{F}{H}=0$ (Definition 2.3). Each one confines the motion to a level set of $F$ — near any point where $\dd F\ne0$, a hypersurface of dimension $2n-1$. With several first integrals $F_1,\dots,F_k$ we collect them into one map, $$F=(F_1,\dots,F_k)\colon\phase\to\R^{k}, \qquad \phase_f=F^{-1}(f)=\{x:F_i(x)=f_i\text{ for all }i\},$$(5.1) and the motion stays in the level set $\phase_f$ through its initial point. Two things can go wrong with such a collection, and each has a name.

Independence at a point is what the implicit function theorem needs: if it holds at $x\in\phase_f$, then near $x$ the level set $\phase_f$ is a smooth submanifold of dimension $2n-k$, with tangent space $$T_x\phase_f=\{v:\dd F_i(v)=0\text{ for all }i\}.$$(5.2) Asking for it only on a dense open set is deliberate: in every example it fails somewhere, and the places where it fails are the interesting ones.

Each of the two conditions is easy to have without the other.

By Proposition 4.11, $n$ is the most one can ask for.

Two remarks on the definition. First, $H=h(F)$ is automatically a first integral of each $F_i$, since it is built from functions that commute with $F_i$; and by the chain rule $\dd H=\sum_i\partial_ih\,\dd F_i$, so $$\hvf{h(F)}=\sum_i\frac{\partial h}{\partial f_i}(F)\,\hvf{F_i}.$$(5.4) Second, the functions are part of the data but not canonical: $(H_1,H_2)$ and $(H,H_2)$ in Example 5.3 define the same level sets.

Deleting any one hypothesis gives a counterexample; three are exercises. The one to see now is compactness, the one the course will not have. The free particle $H=\tfrac12p^{2}$ on $\R^{2}$ is integrable, with $n=1$. At $E>0$ each component of its level set is a line $p=\pm\sqrt{2E}$, and the motion on it is not periodic: this is $T^{0}\times\R^{1}$. When compactness fails but the flows on $\Sigma$ are complete, stage (iv) of the proof shows that a cylinder of this kind is the shape of the conclusion; without completeness there is no such classification. The Toda ring of Lecture 9 is the case the course needs it for.

The proof turns $n$ commuting flows into an action of the group $\R^{n}$ on $\Sigma$, and then identifies $\Sigma$ with a quotient of $\R^{n}$. Groups acting on spaces have not appeared in the course so far, so stage (0) sets up the three words the proof uses.

Two examples fix the words (Figure 5.2). The oscillator flow is an action of $\R$ on the plane; the orbit of a point other than the origin is a circle with stabiliser $(2\pi/\freq)\Z$, while the origin is its own orbit, with stabiliser all of $\R$. And on the torus $\R^{2}/2\pi\Z^{2}$ the flow along the direction $(1,\sqrt2)$ is an action of $\R$ with stabiliser $\{0\}$: the orbit is a copy of $\R$ that never closes, winds densely (Lecture 6), and is still not the whole torus. One flow cannot fill a two-dimensional space; the proof uses $n$ flows on an $n$-dimensional one.

By (I) and the implicit function theorem, near each point of $\Sigma$ the level set is a smooth connected piece of dimension $2n-n=n$, which therefore lies in $\Sigma$; so $\Sigma$ is a smooth $n$-dimensional submanifold with tangent spaces (5.2). The fields $\hvf{F_j}$ are tangent to it, because $$\dd F_i(\hvf{F_j})=\hvf{F_j}F_i=\pb{F_i}{F_j}=0 .$$(5.5) At each point they are linearly independent, since $\iota_{\hvf{F_j}}\symp=\dd F_j$ and $v\mapsto\iota_v\symp$ is injective (3.10). So the $n$ vectors $\hvf{F_1},\dots,\hvf{F_n}$ are a basis of the $n$-dimensional tangent space of $\Sigma$ at every point (Figure 5.3). On that basis $\symp(\hvf{F_i},\hvf{F_j})=\pb{F_i}{F_j}=0$, so $\symp$ vanishes on $T\Sigma$: $\Sigma$ is Lagrangian (Definition 4.10). Spent here: independence, and involution twice.

The fields $\hvf{F_i}$ are tangent to $\Sigma$, so they restrict to vector fields on the compact manifold $\Sigma$, and by Proposition 1.7 they are complete there: each flow $\varphi^{i}_{s}$ is defined on $\Sigma$ for all $s\in\R$ and maps $\Sigma$ to itself. By Corollary 2.6 these flows commute. Define $$\Phi_t=\varphi^{1}_{t_1}\circ\varphi^{2}_{t_2}\circ\cdots\circ\varphi^{n}_{t_n}, \qquad t\in\R^{n}.$$(5.6) Then $\Phi_0=\id$, and $\Phi_{s+t}=\Phi_s\circ\Phi_t$ because the factors commute and each is a one-parameter group. This is an action of $\R^{n}$ on $\Sigma$. Spent here: compactness, for completeness, and involution, for commutativity.

Fix $x_0\in\Sigma$ and consider the orbit map $\Phi^{x_0}\colon\R^{n}\to\Sigma$, $t\mapsto\Phi_t(x_0)$. Since the factors in (5.6) commute, any one of them may be put first, so $$\frac{\partial}{\partial t_i}\Phi_t(x_0)=\hvf{F_i}\big(\Phi_t(x_0)\big).$$ The derivative of $\Phi^{x_0}$ at every $t$ carries the standard basis of $\R^{n}$ to a basis of the tangent space (stage (i)), so by the inverse function theorem $\Phi^{x_0}$ is a local diffeomorphism: it maps a small ball about any $t$ diffeomorphically onto a neighbourhood of $\Phi_t(x_0)$ (Figure 5.4).

Consequently every orbit is open in $\Sigma$. The orbits partition $\Sigma$, so the complement of one orbit is a union of others and is open too; an orbit is therefore open and closed. $\Sigma$ is connected, so there is only one orbit. Spent here: independence (the local diffeomorphism), connectedness, and involution once more (the factors commute).

By Lemma 5.9, with $\Lambda=\Lambda_{x_0}$ now the same for every point of $\Sigma$, $$\R^{n}/\Lambda\;\longrightarrow\;\Sigma, \qquad t\bmod\Lambda\;\longmapsto\;\Phi_t(x_0) \quad\text{is a bijection.}$$(5.7)

This is the step doing the most work, and it has two halves that must not be merged.

$\Lambda$ is discrete. Choose a ball $U$ about $0$ on which $\Phi^{x_0}$ is injective (stage (iii)). If $\lambda\in\Lambda\cap U$ then $\Phi^{x_0}(\lambda)=x_0=\Phi^{x_0}(0)$, so $\lambda=0$. Translating, for any $\lambda\in\Lambda$ the ball $\lambda+U$ contains no other point of $\Lambda$, because $\Lambda-\lambda=\Lambda$. Every point of $\Lambda$ is isolated. Contrast $\Z+\sqrt2\,\Z\subset\R$, a subgroup with points arbitrarily close to $0$: what excludes that here is independence, through the local diffeomorphism.

Discreteness alone gives a cylinder, not a torus. Complete $e_1,\dots,e_k$ to a basis $e_1,\dots,e_n$ of $\R^{n}$. Then $$\R^{k}/\Z^{k}\times\R^{n-k}\;\longrightarrow\;\Sigma, \qquad (\theta,y)\;\longmapsto\; \Phi^{x_0}\Big(\sum_{i\le k}\theta_ie_i+\sum_{j>k}y_je_j\Big)$$ is well defined and bijective by (5.7), and a local diffeomorphism by stage (iii). A bijective local diffeomorphism is a diffeomorphism. So $\Sigma\cong T^{k}\times\R^{n-k}$, and every $k$ from $0$ to $n$ is possible for a discrete stabiliser.

Compactness forces $k=n$. $T^{k}\times\R^{n-k}$ is compact only if $k=n$, and $\Sigma$ is compact. Hence $\Lambda$ has full rank and $\Sigma\cong\R^{n}/\Lambda\cong T^{n}$. Spent here: independence for discreteness, compactness for full rank — the second use of compactness, and the one that fails for the Toda ring of Lecture 9.

With $n=1$ (Figure 5.5) the two cases are a circle, $\Lambda=\tau\Z$ with $\tau$ the period, and a line, $\Lambda=\{0\}$, which is the free particle’s level set.

A point $t\in\R^{n}$ comes back to $x_0$ exactly when $t\in\Lambda$, so the components of $t$ in the basis $e_i$, times $2\pi$, are angles. Write $A=(e_1\;\cdots\;e_n)$ for the matrix whose columns are the basis vectors, and set $$\varphi=2\pi A^{-1}t\pmod{2\pi\Z^{n}}.$$(5.8) These are the angle coordinates of Theorem 5.7: $\varphi\mapsto \Phi_{A\varphi/2\pi}(x_0)$ identifies $\R^{n}/2\pi\Z^{n}$ with $\Sigma$ (Figure 5.6), and along $\gamma_j$ the angle $\varphi_j$ runs once from $0$ to $2\pi$ while the others are constant.

Here is Lecture 3’s annulus again. Each $\varphi_i$ is not a function on $\Sigma$, only a function modulo $2\pi$, but $\dd\varphi_i$ is a well-defined $1$-form, closed since it is locally $\dd$ of a function. It is not exact: $$\oint_{\gamma_j}\dd\varphi_i=2\pi\,\delta_{ij}\ne0\quad(i=j),$$ whereas an exact form integrates to zero round every loop (Proposition 3.10). The torus carries $n$ independent closed non-exact $1$-forms, detected by $n$ loops; Lecture 6 integrates $p\cdot\dd q$ over those same loops.

Let $H=h(F)$. On $\Sigma$ the coefficients in (5.4) are the constants $c_i=\partial_ih(f)$, so $\hvf{H}=\sum_ic_i\hvf{F_i}$, whose flow is $\Phi_{\tau c}$: in $\R^{n}$ the motion is $t\mapsto t+\tau c$. By (5.8), $$\varphi(\tau)=\varphi(0)+\freq\,\tau, \qquad \freq=2\pi A^{-1}\,\nabla h(f).$$(5.9) That is conclusion 3 of Theorem 5.7, and it completes the proof.

One more fact is needed next time, and it is a consequence of stage (i).

So $\oint_\gamma p\cdot\dd q$ over a loop on $\Sigma$ is unchanged when the loop is deformed within $\Sigma$ (Corollary 3.13). That is what Lecture 6 needs to define the action variables $I_i=\frac1{2\pi}\oint_{\gamma_i}p\cdot\dd q$ and to show that, as $f$ varies, the angles (5.8) and the $I_i$ together form canonical coordinates on a whole neighbourhood of $\Sigma$. The theorem as proved here is about one torus at a time; how the tori fit together as $f$ moves is Lecture 6’s business near one torus, and globally it can be subtle (optional lecture A).

By (5.9), with $H=H_1+H_2$: the product lattice and $\nabla h=(1,1)$ give $\freq=(\freq_1,\freq_2)$; the second lattice and $H=F_1$, $\nabla h=(1,0)$, give $\freq=(\freq_1,\freq_2)$ again. The lattice depends on the choice of integrals; the motion does not.

• Tori, or cylinders. If the flows are complete on $\Sigma$ — which compactness guarantees but need not be what supplies it — stages (i)–(iv) give $\Sigma\cong T^{k}\times\R^{n-k}$, and compactness is needed only to force $k=n$. For the Toda ring, whose level sets are not compact, Lecture 9 proves completeness separately and finds $T^{N-1}\times\R$.

• Angles and cycles. The period lattice gives angle coordinates, $n$ basis cycles and a frequency vector (5.9); and, in $\R^{2n}$ or a cotangent bundle, $p\cdot\dd q$ is closed on the torus (Proposition 5.12). Lecture 6 integrates it over the cycles to get the actions, and completes the angles to canonical coordinates.

• Status. Theorem 5.7 is proved modulo the cited Lemma 5.10 and Theorem 1.9. Stage (ii) also uses Corollary 2.6 on $\phase$. On $\R^{2n}$ and on cotangent bundles, where every cotangent chart is canonical (Theorem 4.17), the coordinate proof of Lecture 2 gives it. On a general symplectic manifold it rests on the Jacobi identity for the bracket of $\symp$ and the second proof of the engine, both proved in readings (Lecture 4, Lecture 2).

• Independent on a dense set, or on all of $\Sigma$. The definition allows the integrals to be dependent somewhere; the oscillator’s are, on two planes. The theorem applies only to the components of level sets on which they are independent at every point.

• One torus at a time. The theorem is about one level set. How the tori fit together as $f$ varies is Lecture 6’s business near one torus; globally it can be subtle (optional lecture A).