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SYMPLECTIC VECTOR SPACES, LAGRANGIAN SUBSPACES, AND LIOUVILLE’S THEOREM

The purpose of this paper is to prove Liouville’s theorem on volume-preserving flows from a symplectic linear algebra point of view. Along the way, we will prove a series of properties about symplectic vector spaces that elucidate the usefulness of this approach with regard to the correspondence between the configuration space and the symplectic phase space. Though only briefly discussed, this paper motivates how symplectic manifolds provide the natural abstraction of the phase space of a dynamical system through the simplified linear example: symplectic vector spaces.