Symplectic vector spaces and symplectic vector bundles

Abstract

This chapter is divided into three parts. The first and most substantial of these is devoted to symplectic vector spaces. Using methods of exterior algebra which are due to Elie Cartan, we study the rank of an exterior 2-form on a real vector space as well as that of the 2-form induced on a vector subspace. By means of associated canonical bases we show that there is only one model of a symplectic form and that vector subspaces may be classified by the rank of the induced form. In mechanics one uses mainly isotropic and coisotropic subspaces, particularly Lagrangian subspaces which will be discussed here in some detail. In particular we study the reduction of a symplectic vector space.