Lagrangian submanifolds and action–angle coordinates

In a symplectic vector space of dimension 2n, a subspace is isotropic when the form vanishes on every pair of its vectors. Such a subspace has dimension at most n. It is Lagrangian when its dimension is exactly n, equivalently when it equals its symplectic orthogonal. A smooth embedded submanifold L is Lagrangian when every tangent space has this property, or when dim L=n and the pulled-back form is zero. Checking only the zero pullback is insufficient: a curve in a four-dimensional phase space is automatically isotropic but has too small a dimension to be Lagrangian.

In T*Q, the zero section and each fiber are Lagrangian by direct canonical-coordinate calculation. For the graph of a one-form β, pullback of θ is β and pullback of ω is −dβ. Thus the graph is Lagrangian exactly when β is closed. Locally a closed β is dS, so a local scalar potential describes the graph. Globally, closedness does not imply one single-valued potential; the circle translation example persists. A Lagrangian condition is a geometric restriction, while exactness adds global information about one-form periods.

For H=(q²+p²)/2 on the punctured phase plane, write q=√(2I)sin θ and p=√(2I)cos θ, with I>0 and θ modulo 2π. Direct differentiation gives dq∧dp=dθ∧dI. This angular orientation matters: our Hamilton equations give θ̇=1 and Ī=0, matching q̇=p,ṗ=−q. The action is one over 2π times the integral of θ_canonical=p dq around the orbit in the direction of the flow. The same symbol θ often names both a one-form and an angle, so we identify which meaning is intended in each formula.

For the oscillator, the integral is the area π times the squared radius, equal to 2πI, so the normalized action is I=H. The angle coordinate is periodic rather than a global real number, and the transformation is singular at I=0 where the whole circle collapses to an equilibrium. Consequently no action–angle chart of this type covers the origin. The energy circle is a one-dimensional Lagrangian submanifold of the plane, but its neighboring circles are needed to supply the full two-dimensional coordinate system.

A single Lagrangian submanifold need not be invariant under a given Hamiltonian. Invariance requires X_H to be tangent; for a connected Lagrangian L this is equivalent to H restricted to L being constant. The equivalence follows because ω(X_H,v)=dH(v) and the symplectic orthogonal of TL equals TL. Even an invariant Lagrangian graph does not automatically provide a neighborhood filled by compact invariant tori. The action–angle theorem requires enough commuting independent functions plus a compact connected regular joint fiber.

Lagrangian graphs also appear in Hamilton–Jacobi theory: a potential S solving H(q,dS)=E makes its graph lie in one energy level, and hence invariant wherever it is a smooth Lagrangian graph. Caustics can make projection to configuration space fail, although the phase-space submanifold remains smooth. A single generating potential may then require several charts. This explains why a geometrically simple phase-space object can acquire multivalued momentum branches when viewed only through position coordinates.