Symplectic manifold

A manifold $M$ together with a symplectic form. Classical example: cotangent bundle of a configuration space with the Poincare 2-form (see Hamiltonian mechanics).

They give rise to classical Hamiltonian systems.

We can define an associated Poisson bracket in $M$, so it is a particular case of Poisson manifold whose Poisson bracket is of maximal rank. It can be proven that symplectic manifolds are even-dimensional: the Poisson bracket is a skew-symmetric matrix, so the rank is always even.

In the finite dimensional case we can use, I think, the Pfaff-Darboux theorem to obtain coordinates which provide a particularly simple expression for the symplectic form. According to @olver86 page 390 it is not true in the infinite dimensional case.

Example
In one-dimensional classical mechanics, the phase space is two-dimensional and can be parameterized by a position variable $q$ and a momentum variable $p$. The symplectic 2-form in this phase space is defined as:

$◼$

The important transformations between symplectic manifolds are called symplectomorphisms or canonical transformations.
There are two important Lie subalgebras of $\mathfrak{X}(M)$:

• symplectic vector fields
• Hamiltonian vector fields. This is a subalgebra of the previous one.