Symplectic Manifolds

A symplectic manifold is a mathematical object studied in the field of differential geometry, and it has important applications in classical mechanics and mathematical physics, particularly in Hamiltonian mechanics and the study of phase spaces.

Definition 1. A symplectic manifold $M$ is a smooth manifold $M$ equipped with a closed, non-degenerate 2-form $\omega$, called the symplectic form. A symplectic manifold has an even dimension, as the non-degeneracy condition requires it. In the context of classical mechanics, a symplectic manifold represents the phase space of a mechanical system, where the symplectic form corresponds to the Poisson bracket, a fundamental structure that encodes the dynamics of the system.

Here's a quick overview of the terms involved:

1. Definition 2. Smooth manifold: A smooth manifold is a topological space that locally looks like Euclidean space (in the sense that it is homeomorphic to an open subset of Euclidean space) and has a smooth structure, which is a maximal atlas of smooth coordinate charts. These charts are compatible with each other in the sense that the transition maps between charts are smooth functions.

2. Definition 3. 2-form: A 2-form is a differential 2-form, which is a smooth, alternating, and multilinear map that takes two tangent vectors and returns a scalar. It's a generalization of the notion of a surface element in Euclidean space.

3. Definition 4. Closed: A 2-form $\omega$ is closed if its exterior derivative, $d\omega$, is zero. The exterior derivative is an operation that generalizes the concept of a differential in calculus.

4. Definition 5. Non-degenerate: A 2-form $\omega$ is non-degenerate if, for every non-zero tangent vector $v$ at a point in the manifold, there exists another tangent vector $w$ such that $\omega(v, w) \neq 0$.