Symplectic manifolds constitute the mathematical structure for modern Hamiltonian mechanics. Symplectic manifolds can also be seen as even dimensional analogues to contact manifolds.

Properties

1. Every symplectic manifold is even dimensional. This is easy to understand in view of the physics. In Hamilton equations, location and momentum vectors always appear in pairs.
2. A form $\omega\in \Omega^2(M)$ on a $2n$ -dimensional manifold $M$ is non-degenerate if and only if the $n$ -fold product $\omega^n= \omega\wedge \cdots \wedge \omega$ is non-zero.
3. As a consequence of the last property, every symplectic manifold is orientable.

Let $(M,\omega)$ and $(N,\eta)$ be symplectic manifolds. Then a diffeomorphism $f\colon M\to N$ is called a symplectomorphism if $f^*\eta=\omega$ , that is, if the symplectic form on $N$ pulls back to the form on $M$ .

Notes