Symplectic Geometry from a Dynamical Systems Point of View

The easiest example of a symplectic manifold is the standard sym-plectic vector space V := R 2n with coordinates (q 1 ; p 1 ; :::; q n ; p n) and symplectic form ! = n X j=1 dq j ^ dp j : The basic feature of ! is that it is a closed non-degenerate 2-form on R 2n. In particular it gives a natural indentiication of V with its dual V v ! !(v; :): As an auxiliary structure we also use the standard inner product h:; :i = n X j=1 dq j dq j + dp j dp j ]: The structure preserving maps will be the smooth maps : U ! V , where U V is open and ! = !jU. They are called symplectic maps. Clearly such a map is locally a diieomorphism. These maps arise originally in the study of Hamiltonian mechanics, where the Hamilton equations _ p j = ? @H @q j ; _ q j = @H @p j produce ows consisting of symplectic maps. We can deene a symplectic manifold in the following way. Assume that M is a topological manifold of even dimension. An atlas A on M is said to be symplectic if the transition maps are symplectic as maps between open subsets of V. Two symplectic atlases are equivalent if the union is a symplectic atlas. A maximal symplectic atlas S is called