The Lagrangian in Symplectic Mechanics

Abstract

In this paper we argue that the symplectic description of classical mechanics contains many elements of the Lagrange formulation of classical mechanics, in particular a variational description in terms of an action functional. In order to obtain these results, one has to enlarge the symplectic formulation with part of the prequantization construction: the construction of a principal S 1-bundle with connection over the symplectic manifold, but without the quantization condition. We propose to call this enlargement the ‘postclassical formalism.’ Apart from mathematical evidence that the post-classical formalism is a (very) useful enlargement of symplectic mechanics, we also argue that this formalism has physical content.