Symplectic Manifold

A pair $(M,\omega)$, where $M$ is a Manifold and $\omega$ is a Symplectic Form on $M$. The Phase Space $\Bbb{R}^{2n}=\Bbb{R}^n\times\Bbb{R}^n$ is a symplectic manifold. Near every point on a symplectic manifold, it is possible to find a set of local ``Darboux coordinates'' in which the Symplectic Form has the simple form

$$\omega=\sum_k dq_k\wedge dp_k$$

(Sjamaar 1996), where $dq_k\wedge dp_k$ is a Wedge Product.

See also Manifold, Symplectic Diffeomorphism, Symplectic Form

References

Sjamaar, R. ``Symplectic Reduction and Riemann-Roch Formulas for Multiplicities.'' Bull. Amer. Math. Soc. 33, 327-338, 1996.