Symplectic Vector Spaces

DefinitionSymplectic Form

A symplectic form on a real vector space $V$ is a bilinear map $\omega: V \times V \to \mathbb{R}$ satisfying:

Skew-symmetry: $\omega(v, w) = -\omega(w, v)$ for all $v, w \in V$
Non-degeneracy: If $\omega(v, w) = 0$ for all $w \in V$ , then $v = 0$

A vector space equipped with a symplectic form is called a symplectic vector space $(V, \omega)$ .

The symplectic form generalizes the notion of area in classical mechanics. Unlike inner products, symplectic forms are antisymmetric rather than symmetric, capturing the oriented nature of phase space in Hamiltonian mechanics.

ExampleStandard Symplectic Form on $\mathbb{R}^{2n}$

The standard symplectic form on $\mathbb{R}^{2n}$ with coordinates $(q_1, \ldots, q_n, p_1, \ldots, p_n)$ is:

$\omega_0 = \sum_{i=1}^n dq_i \wedge dp_i$

In matrix notation, if $v = (q, p)$ and $w = (q', p')$ , then:

$\omega_0(v, w) = \sum_{i=1}^n (q_i p_i' - p_i q_i') = v^T J w$

where $J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}$ is the standard symplectic matrix.

Remark

Every symplectic vector space has even dimension. This follows from the non-degeneracy condition: the matrix representing $\omega$ in any basis must be invertible and antisymmetric, which implies the dimension must be even.

The non-degeneracy of $\omega$ establishes an isomorphism between $V$ and its dual $V^*$ via $v \mapsto \omega(v, \cdot)$ . This allows us to "raise and lower indices" similar to Riemannian geometry, but with crucial differences due to the antisymmetric nature of the form.

DefinitionSymplectic Complement

For a subspace $W \subseteq V$ , the symplectic complement is:

$W^\omega = \{v \in V : \omega(v, w) = 0 \text{ for all } w \in W\}$

Unlike orthogonal complements, we may have $W \cap W^\omega \neq \{0\}$ .

ExampleIsotropic and Lagrangian Subspaces

A subspace $W$ is:

Isotropic if $W \subseteq W^\omega$ (i.e., $\omega|_W = 0$ )
Coisotropic if $W^\omega \subseteq W$
Lagrangian if $W = W^\omega$ (maximal isotropic)

In $(\mathbb{R}^{2n}, \omega_0)$ , the subspace $\{(q, 0) : q \in \mathbb{R}^n\}$ spanned by position coordinates is Lagrangian.

Lagrangian subspaces play a fundamental role in symplectic geometry, analogous to the role of totally geodesic submanifolds in Riemannian geometry. They have dimension exactly $n = \frac{1}{2}\dim V$ and represent maximal subspaces on which the symplectic form vanishes.