05.09.E1 · symplectic / integrable

Symplectic geometry and integrable systems exercise pack (Arnold Part III appendices supplement)

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Formal definition of the pack Intermediate

A symplectic manifold $(M^{2 n}, ω)$ carries a closed nondegenerate 2-form. Its distinguished submanifolds are the Lagrangians — maximal isotropics of dimension $n$ , on which $ω$ restricts to zero. A Hamiltonian action of a Lie group $G$ has a moment map $μ : M \to g^{*}$ packaging the conserved quantities, the geometric form of Noether's theorem. The Liouville-Arnold theorem closes the circle: if $n$ functions Poisson-commute and are independent on a compact connected level set, that level set is an $n$ -torus, and a neighbourhood carries action-angle coordinates $(I, θ)$ in which the flow is linear. Arnold's Part III appendices treat these as a single package — symplectic structure, moment map, integrability.

This pack collects nine exercises — two easy, four medium, three hard — each with a hint and a full solution. It is meant to be read alongside its prerequisite units 05.01.02, 05.02.03, 05.02.04, 05.04.01, and 05.05.01, not as a standalone development. The problems are grouped by topic: checking symplectic forms and Lagrangian submanifolds (easy/medium), constructing moment maps (medium), and applying Liouville-Arnold to produce action-angle variables (hard).

The conventions throughout are Arnold's and Cannas's: $(M, ω)$ symplectic with $ω^{n} \neq = 0$ ; a Lagrangian $L$ satisfies $dim L = n$ and $ω ∣_{L} = 0$ ; the moment map obeys $ι_{X_{ξ}} ω = d ⟨ μ, ξ ⟩$ for each $ξ \in g$ ; action variables are $I_{j} = \frac{1}{2 π} \oint_{γ_{j}} θ$ over the basis cycles $γ_{j}$ of the Liouville torus, with $θ = p d q$ .

Key theorem with full solution Intermediate

Before the pack proper, we work one exercise in full as an exemplar of the format. The remaining eight follow the same structure (problem, hint, full answer in <details> blocks).

Lead exercise. State and prove the torus half of the Liouville-Arnold theorem: a compact connected regular level set of $n$ independent Poisson-commuting integrals on $(M^{2 n}, ω)$ is diffeomorphic to the $n$ -torus $T^{n}$ .

Solution. Let $f_{1}, \dots, f_{n} : M \to R$ satisfy ${f_{i}, f_{j}} = 0$ for all $i, j$ , with $d f_{1}, \dots, d f_{n}$ linearly independent on the level set $M_{c} = {f = c}$ , which we assume compact and connected. Set $X_{i} = X_{f_{i}}$ , the Hamiltonian vector fields.

The $X_{i}$ are tangent to $M_{c}$ . Since $X_{i} (f_{j}) = {f_{j}, f_{i}} = 0$ , each $X_{i}$ preserves every $f_{j}$ , so it is tangent to $M_{c}$ .

The $X_{i}$ commute. The Lie bracket of Hamiltonian fields is $[X_{i}, X_{j}] = - X_{{f_{i}, f_{j}}} = X_{0} = 0$ because ${f_{i}, f_{j}} = 0$ . So the flows commute.

The $X_{i}$ are independent and span $T M_{c}$ . Nondegeneracy of $ω$ and independence of the $d f_{i}$ force $X_{1}, \dots, X_{n}$ to be pointwise linearly independent; since $dim M_{c} = 2 n - n = n$ , they frame $T M_{c}$ .

So $M_{c}$ carries $n$ commuting, complete (by compactness), pointwise-independent vector fields. Their joint flow defines an action of $R^{n}$ on $M_{c}$ :

Φ : R^{n} \times M_{c} \to M_{c}, Φ (t, x) = g_{X_{1}}^{t_{1}} \dots g_{X_{n}}^{t_{n}} (x) .

This action is transitive (the orbit of any point is open and closed in connected $M_{c}$ ) and locally free. The stabiliser of a point is a discrete subgroup — a lattice $Λ \subset R^{n}$ — and is the same lattice everywhere by transitivity. Compactness forces $Λ$ to have full rank $n$ . Therefore

M_{c} ≅ R^{n} /Λ ≅ T^{n},

the $n$ -torus. $□$

This is Arnold's signature theorem: integrability forces the regular compact level sets to be tori, foliating phase space, with the flow becoming linear (quasi-periodic) on each torus. The action-angle coordinates of the next exercises are the global completion of this picture.

Exercises Intermediate

Exercise 1 (easy, symbolic). The standard form on $R^{2 n}$ is symplectic.

Verify that $ω = \sum_{i = 1}^{n} d p_{i} \land d q_{i}$ on $R^{2 n}$ is closed and nondegenerate.

Hint

Closedness: $ω$ has constant coefficients. Nondegeneracy: compute $ω^{n}$ and check it is a nonzero multiple of the volume form.

Answer

Closed. $ω = \sum_{i} d p_{i} \land d q_{i}$ has constant coefficients, so $d ω = \sum_{i} d (d p_{i} \land d q_{i}) = 0$ since $d (d p_{i}) = d (d q_{i}) = 0$ .

Nondegenerate. Compute the top power. Because the $d p_{i} \land d q_{i}$ for distinct $i$ commute as 2-forms and each squares to zero,

ω^{n} = (i \sum d p_{i} \land d q_{i})^{n} = n! d p_{1} \land d q_{1} \land \dots \land d p_{n} \land d q_{n},

which is $n!$ times the standard volume form, hence nowhere zero. A 2-form with $ω^{n} \neq = 0$ is nondegenerate. So $(R^{2 n}, ω)$ is symplectic — the Darboux model every symplectic manifold looks like locally. Unit 05.01.02.

Exercise 2 (easy, proof). The graph of $df$ is Lagrangian.

Let $f : M \to R$ be smooth and $T^{*} M$ its cotangent bundle with $ω = - d θ$ . Show the graph ${(q, d f_{q}) : q \in M}$ is a Lagrangian submanifold of $T^{*} M$ .

Hint

The graph is the image of the section $s = df : M \to T^{*} M$ . Pull back $θ$ by this section: $s^{*} θ = df$ .

Answer

The graph $Γ_{df}$ is the image of the 1-form $df$ viewed as a section $s : M \to T^{*} M$ , $s (q) = (q, d f_{q})$ . Its dimension is $dim M = n$ , half of $dim T^{*} M = 2 n$ — the right dimension for a Lagrangian.

The tautological 1-form $θ$ has the reproducing property $s^{*} θ = α$ for any 1-form $α = s$ . Here $s = df$ , so $s^{*} θ = df$ . Therefore

s^{*} ω = s^{*} (- d θ) = - d (s^{*} θ) = - d (df) = - d^{2} f = 0.

So $ω$ restricts to zero on $Γ_{df}$ , and with the half-dimension count this makes $Γ_{df}$ Lagrangian. (More generally, the graph of a closed 1-form is Lagrangian, and a section is Lagrangian iff the 1-form is closed.) Unit 05.05.01.

Exercise 3 (medium, symbolic). Moment map for the rotation $S^{1}$ -action on $R^{2}$ .

Let $S^{1}$ act on $(R^{2}, ω = d p \land d q)$ by rotation. Find a moment map $μ : R^{2} \to R ≅ u (1)^{*}$ .

Hint

The generating vector field of rotation is $X = q \partial_{p} - p \partial_{q}$ . Solve $ι_{X} ω = d μ$ .

Answer

The $S^{1}$ -rotation by angle $θ$ generates the vector field $X = - p \partial_{q} + q \partial_{p}$ (infinitesimal rotation). Contract with $ω = d p \land d q$ :

ι_{X} ω = ι_{X} (d p \land d q) = (ι_{X} d p) d q - (ι_{X} d q) d p = q d q - (- p) d p = q d q + p d p .

We need $μ$ with $ι_{X} ω = d μ$ :

d μ = q d q + p d p = d (\frac{1}{2} (q^{2} + p^{2})) .

So $μ = \frac{1}{2} (p^{2} + q^{2})$ is a moment map (up to an additive constant). Geometrically it is the harmonic-oscillator Hamiltonian — the conserved quantity (here the energy / the radial action) for rotational symmetry, exactly the Noether quantity the moment map packages. Arnold Appendix 5; unit 05.04.01.

Exercise 4 (medium, proof). Equivariance of the moment map for a $G$ -action.

For a Hamiltonian action of $G$ with moment map $μ : M \to g^{*}$ , suppose $μ$ is $G$ -equivariant for the coadjoint action. Show that the components $μ^{ξ} := ⟨ μ, ξ ⟩$ satisfy ${μ^{ξ}, μ^{η}} = μ^{[ξ, η]}$ .

Hint

Differentiate the equivariance relation, or compute ${μ^{ξ}, μ^{η}} = ω (X_{ξ}, X_{η})$ and use that the assignment $ξ \mapsto X_{ξ}$ is a Lie-algebra antihomomorphism (or homomorphism, per convention).

Answer

By definition of the moment map, $X_{ξ} = X_{μ^{ξ}}$ is the Hamiltonian vector field of $μ^{ξ}$ . The Poisson bracket of two Hamiltonians equals $ω$ applied to their fields:

{μ^{ξ}, μ^{η}} = ω (X_{μ^{ξ}}, X_{μ^{η}}) = ω (X_{ξ}, X_{η}) .

The infinitesimal action is a Lie-algebra homomorphism $ξ \mapsto X_{ξ}$ with $[X_{ξ}, X_{η}] = X_{[ξ, η]}$ . The bracket of Hamiltonian fields satisfies $[X_{f}, X_{g}] = X_{{f, g}}$ , so comparing,

X_{{μ^{ξ}, μ^{η}}} = [X_{μ^{ξ}}, X_{μ^{η}}] = [X_{ξ}, X_{η}] = X_{[ξ, η]} = X_{μ^{[ξ, η]}} .

Two Hamiltonians with the same Hamiltonian vector field differ by a locally constant function; equivariance pins this constant to zero, giving

{μ^{ξ}, μ^{η}} = μ^{[ξ, η]} .

So the moment map is a Poisson map intertwining the Lie bracket on $g$ with the Poisson bracket on $M$ — the bracket-preserving property that makes equivariant moment maps the geometric incarnation of Noether's theorem. Arnold Appendix 5; unit 05.04.01.

Exercise 5 (medium, numeric). Action variable as enclosed area for the oscillator.

For $H = \frac{1}{2} (p^{2} + ω_{0}^{2} q^{2})$ , compute the action variable $I = \frac{1}{2 π} \oint p d q$ on the energy level $H = E$ and confirm $H = ω_{0} I$ .

Hint

The level set is an ellipse. The loop integral $\oint p d q$ equals its enclosed area.

Answer

The level set $H = E$ is the ellipse $\frac{1}{2} p^{2} + \frac{1}{2} ω_{0}^{2} q^{2} = E$ , i.e. $\frac{p ^{2}}{2 E} + \frac{q ^{2}}{2 E / ω _{0}^{2}} = 1$ , with semi-axes $a = 2 E$ (in $p$ ) and $b = 2 E / ω_{0}$ (in $q$ ). By Stokes/Green's theorem, $\oint p d q$ equals the enclosed area:

\oint p d q = π a b = π \cdot 2 E \cdot \frac{2 E}{ω _{0}} = \frac{2 π E}{ω _{0}} .

Therefore

I = \frac{1}{2 π} \oint p d q = \frac{1}{2 π} \cdot \frac{2 π E}{ω _{0}} = \frac{E}{ω _{0}} .

Inverting, $E = ω_{0} I$ , so $H = ω_{0} I$ in action variables. The angular frequency is $\partial H / \partial I = ω_{0}$ , independent of amplitude — the isochronism of the harmonic oscillator. This is the simplest action-angle computation, Arnold §50; unit 05.02.04.

Exercise 6 (medium, symbolic). Poisson-commuting integrals for a separable system.

For $H = \frac{1}{2} (p_{1}^{2} + p_{2}^{2}) + V_{1} (q_{1}) + V_{2} (q_{2})$ on $R^{4}$ , exhibit two independent Poisson-commuting integrals and conclude the system is Liouville-integrable.

Hint

The energy in each separate degree of freedom is conserved. Check the bracket of $f_{1} = \frac{1}{2} p_{1}^{2} + V_{1} (q_{1})$ and $f_{2} = \frac{1}{2} p_{2}^{2} + V_{2} (q_{2})$ .

Answer

Take

f_{1} = \frac{1}{2} p_{1}^{2} + V_{1} (q_{1}), f_{2} = \frac{1}{2} p_{2}^{2} + V_{2} (q_{2}),

so $H = f_{1} + f_{2}$ . Each $f_{i}$ depends only on the $i$ -th pair $(q_{i}, p_{i})$ . Their Poisson bracket in the four-dimensional phase space is

{f_{1}, f_{2}} = i = 1 \sum 2 (\partial_{q_{i}} f_{1} \partial_{p_{i}} f_{2} - \partial_{p_{i}} f_{1} \partial_{q_{i}} f_{2}) = 0,

because $f_{1}$ has no $(q_{2}, p_{2})$ -dependence and $f_{2}$ has no $(q_{1}, p_{1})$ -dependence: every term contains a factor that vanishes. They are independent wherever $p_{1}, p_{2}$ are not both forced equal by the level constraints (generically $d f_{1}, d f_{2}$ are linearly independent). With $n = 2$ Poisson-commuting independent integrals on a $4$ -dimensional symplectic manifold, the system is Liouville-integrable, and on compact connected level sets the Liouville-Arnold theorem gives $T^{2}$ tori with action-angle coordinates. Units 05.02.03, 05.02.04.

Exercise 7 (hard, proof). Liouville tori are Lagrangian.

In the setting of the Liouville-Arnold theorem, show that each regular level set $M_{c} = {f_{1} = c_{1}, \dots, f_{n} = c_{n}}$ is a Lagrangian submanifold of $(M^{2 n}, ω)$ .

Hint

$M_{c}$ has dimension $n$ . It remains to show $ω ∣_{M_{c}} = 0$ . The tangent space is spanned by the Hamiltonian fields $X_{f_{i}}$ ; compute $ω (X_{f_{i}}, X_{f_{j}})$ .

Answer

The level set $M_{c}$ is cut out by $n$ functions with independent differentials, so $dim M_{c} = 2 n - n = n$ — half-dimensional, the dimension count for a Lagrangian.

As shown in the lead exercise, the Hamiltonian vector fields $X_{f_{1}}, \dots, X_{f_{n}}$ are tangent to $M_{c}$ and pointwise independent, so they frame the tangent space $T_{x} M_{c}$ at each point. Evaluate $ω$ on this frame:

ω (X_{f_{i}}, X_{f_{j}}) = {f_{i}, f_{j}} = 0

for all $i, j$ , by the defining Poisson-commutativity. Since the $X_{f_{i}}$ span $T_{x} M_{c}$ , this says $ω ∣_{T_{x} M_{c}} = 0$ : the form restricts to zero on the level set.

A half-dimensional submanifold on which $ω$ vanishes is Lagrangian. So each compact regular Liouville torus is a Lagrangian $T^{n} \subset M$ . The invariant tori of an integrable system foliate phase space by Lagrangians — this is the geometric content tying 05.05.01, 05.02.03, and 05.02.04 together, and the configuration KAM perturbs. Arnold §49.

Exercise 8 (hard, symbolic). Action-angle variables linearise the flow.

Assume action-angle coordinates $(I, θ)$ exist on a Liouville torus with $ω = \sum_{j} d I_{j} \land d θ_{j}$ and $H = H (I)$ alone. Solve Hamilton's equations and show the motion is quasi-periodic on the torus.

Hint

Write Hamilton's equations for $H = H (I)$ in the $(I, θ)$ chart. Since $H$ is independent of $θ$ , the actions are constant.

Answer

In action-angle coordinates with $ω = \sum_{j} d I_{j} \land d θ_{j}$ , Hamilton's equations are

\dot{I}_{j} = - \frac{\partial H}{\partial θ _{j}}, \dot{θ}_{j} = \frac{\partial H}{\partial I _{j}} .

Because $H = H (I)$ does not depend on the angles, $\partial H / \partial θ_{j} = 0$ , so

\dot{I}_{j} = 0 \Rightarrow I_{j} = const, \dot{θ}_{j} = \frac{\partial H}{\partial I _{j}} (I) =: ν_{j} (I) = const .

The frequencies $ν_{j}$ are constant because they depend only on the (constant) actions. Integrating,

I_{j} (t) = I_{j} (0), θ_{j} (t) = θ_{j} (0) + ν_{j} t (mod 2 π) .

The motion stays on the torus ${I = const}$ and winds around it at constant angular frequencies $ν_{j} (I)$ — quasi-periodic (conditionally periodic) motion. It is periodic when the $ν_{j}$ are commensurate and densely fills the torus when they are rationally independent. This linear-flow-on-a-torus is the canonical form of every integrable system near a regular compact level set, and the unperturbed picture KAM theory studies 05.09.01. Arnold §50; unit 05.02.04.

Exercise 9 (hard, proof). Darboux coordinates exist locally — uniqueness of the local model.

Argue that every point of a symplectic manifold $(M^{2 n}, ω)$ has a chart in which $ω = \sum_{i} d p_{i} \land d q_{i}$ , so all symplectic manifolds of a given dimension are locally indistinguishable. Sketch the Moser-trick proof.

Hint

Linear algebra gives a basis at one point where $ω$ is standard; the Moser trick (Moser's homotopy method) upgrades this to a neighbourhood by absorbing the difference $ω_{t} = ω_{0} + t (ω_{1} - ω_{0})$ into a flow.

Answer

Pointwise normal form. By the linear theory, the antisymmetric nondegenerate form $ω_{x}$ on $T_{x} M$ admits a symplectic basis, so in suitable linear coordinates $ω$ agrees with the standard form $ω_{0} = \sum_{i} d p_{i} \land d q_{i}$ at the single point $x$ .

Moser trick to a neighbourhood. On a ball around $x$ , interpolate $ω_{t} = ω_{0} + t (ω - ω_{0})$ for $t \in [0, 1]$ . Each $ω_{t}$ is closed, and (shrinking the ball) nondegenerate since the family starts and stays near $ω_{0}$ . By the Poincaré lemma the closed form $ω - ω_{0}$ , which vanishes at $x$ , is exact: $ω - ω_{0} = d σ$ with $σ$ vanishing to first order at $x$ . Define the time-dependent vector field $X_{t}$ by $ι_{X_{t}} ω_{t} = - σ$ (solvable by nondegeneracy). Its flow $ψ_{t}$ satisfies, via Cartan's formula,

\frac{d}{d t} ψ_{t}^{*} ω_{t} = ψ_{t}^{*} (L_{X_{t}} ω_{t} + \frac{d}{d t} ω_{t}) = ψ_{t}^{*} (d ι_{X_{t}} ω_{t} + (ω - ω_{0})) = ψ_{t}^{*} (- d σ + d σ) = 0.

So $ψ_{1}^{*} ω = ψ_{1}^{*} ω_{1} = ψ_{0}^{*} ω_{0} = ω_{0}$ . The diffeomorphism $ψ_{1}$ is the Darboux chart: it pulls $ω$ back to the standard form. Hence symplectic manifolds have no local invariants beyond dimension — the contrast with Riemannian geometry, where curvature obstructs local flatness. Units 05.01.02, and the Moser trick of 05.01.05.

Exercise pack EP. Arnold Mathematical Methods of Classical Mechanics Part III appendices supplement: symplectic forms, Lagrangian submanifolds, moment maps, action-angle variables, and the Liouville-Arnold integrability theorem across §49-§50 and Appendices 3, 5.

Prerequisites

05.01.02 pending
05.02.03 pending
05.02.04 pending
05.04.01 pending
05.05.01 pending

Tier anchors

beginner
intermediate: Arnold *Mathematical Methods of Classical Mechanics* Part III + appendices exercises (§49-§50: Liouville-Arnold theorem, action-angle; Appendix 5: symmetry and the moment map); Cannas da Silva *Lectures on Symplectic Geometry* §22-§27
master

References

Arnold — Mathematical Methods of Classical Mechanics (2nd ed., GTM 60) · Ch. 10 §49-§50 (Liouville-Arnold theorem, action-angle variables) and Appendices 3, 5 (symplectic structures, dynamical systems with symmetry and the moment map)
Cannas da Silva — Lectures on Symplectic Geometry · §22-§27 — symplectic forms, moment maps, Marsden-Weinstein reduction, Liouville-Arnold and action-angle
McDuff-Salamon — Introduction to Symplectic Topology (3rd ed.) · Ch. 3-5 — symplectic linear algebra, symplectomorphisms, moment maps and Hamiltonian group actions

Estimated time

intermediate: 5h