Symplectic Submanifolds and Lagrangians

DefinitionSymplectic Submanifold

A submanifold $N \subseteq (M, \omega)$ is symplectic if the restriction $\omega|_N$ is a symplectic form on $N$ . This requires:

$\omega|_N$ is non-degenerate on $T_pN$ for all $p \in N$
$\dim N$ is even

A symplectic submanifold inherits a symplectic structure from the ambient manifold.

Symplectic submanifolds are relatively rare compared to arbitrary submanifolds. The non-degeneracy condition is restrictive: not every even-dimensional submanifold carries a symplectic structure.

ExampleSymplectic Submanifolds in $\mathbb{R}^{2n}$

Consider $\mathbb{R}^{2n}$ with standard symplectic form $\omega_0 = \sum_{i=1}^n dq_i \wedge dp_i$ . The submanifold:

$N = \{(q_1, \ldots, q_k, 0, \ldots, 0, p_1, \ldots, p_k, 0, \ldots, 0) : q_i, p_i \in \mathbb{R}\} \cong \mathbb{R}^{2k}$

is symplectic with $\omega|_N = \sum_{i=1}^k dq_i \wedge dp_i$ . This is the standard embedding of lower-dimensional phase spaces.

DefinitionLagrangian Submanifold

A submanifold $L \subseteq (M, \omega)$ is Lagrangian if:

$\omega|_L = 0$ (i.e., $L$ is isotropic)
$\dim L = \frac{1}{2}\dim M$ (i.e., $L$ is maximal isotropic)

Equivalently, at each $p \in L$ , we have $T_pL = (T_pL)^\omega$ in the symplectic vector space $(T_pM, \omega_p)$ .

ExampleZero Section and Fiber

In the cotangent bundle $T^*Q$ with canonical symplectic form $\omega = -d\theta$ :

The zero section $Q \times \{0\} \subseteq T^*Q$ is Lagrangian
Each cotangent fiber $T^*_qQ = \{q\} \times T^*_qQ$ is Lagrangian

These provide the fundamental examples of Lagrangian submanifolds in the phase space of classical mechanics.

Remark

Lagrangian submanifolds are the "largest" submanifolds on which $\omega$ vanishes. They cannot be increased in dimension while maintaining isotropy, and they have exactly half the dimension of the ambient space.

The study of Lagrangian submanifolds is central to symplectic topology. They arise naturally as fixed-point loci, as graphs of canonical transformations, and as quantum states in geometric quantization.

DefinitionWeinstein Lagrangian Neighborhood Theorem

For any Lagrangian submanifold $L \subseteq (M, \omega)$ , there exists a neighborhood $U$ of $L$ symplectomorphic to a neighborhood of the zero section in $T^*L$ .

This means every Lagrangian looks locally like the zero section of its own cotangent bundle.

ExampleLagrangians from Generating Functions

Let $S: Q \to \mathbb{R}$ be a smooth function on a manifold $Q$ . The graph:

$L_S = \{(q, dS_q) : q \in Q\} \subseteq T^*Q$

is a Lagrangian submanifold. The 1-form $dS$ embeds $Q$ as a Lagrangian in its cotangent bundle. Different choices of $S$ give different Lagrangian sections.

DefinitionCoisotropic Submanifold

A submanifold $C \subseteq (M, \omega)$ is coisotropic if:

$(T_pC)^\omega \subseteq T_pC$

for all $p \in C$ . Equivalently, the symplectic complement of the tangent space lies within the tangent space.

Remark

The classification of submanifolds by symplectic type:

Symplectic: $\omega|_N$ non-degenerate, $(T_pN)^\omega \cap T_pN = \{0\}$
Isotropic: $\omega|_N = 0$ , $T_pN \subseteq (T_pN)^\omega$
Lagrangian: $\omega|_N = 0$ and $\dim N = \frac{1}{2}\dim M$ , $T_pN = (T_pN)^\omega$
Coisotropic: $(T_pN)^\omega \subseteq T_pN$

These form a hierarchy of increasingly special submanifolds.

ExampleConstraint Surfaces

In classical mechanics with constraints, the constraint surface is typically coisotropic. For instance, if $C = \{H = 0\}$ is an energy surface in phase space, then $C$ is coisotropic because the Hamiltonian vector field $X_H$ is tangent to $C$ , and the characteristic foliation corresponds to the dynamics.

DefinitionConormal Bundle

For a submanifold $N \subseteq Q$ , the conormal bundle in $T^*Q$ is:

$N^* = \{(q, \alpha) \in T^*Q : q \in N, \alpha|_{T_qN} = 0\}$

This is always a Lagrangian submanifold of $T^*Q$ , even when $N$ has no special structure in $Q$ .