A variational principle for actions on symmetric symplectic spaces

TL;DR

This paper introduces a variational principle for actions on symmetric symplectic spaces, providing a geometric framework for generating functions, composition of actions, and classical trajectories, with explicit treatment of simple cases.

Contribution

It defines generating functions for canonical relations on symmetric symplectic spaces and links them to Hamiltonians through a new variational principle, expanding geometric understanding.

Findings

Actions compose via a geometric formula

The variational principle determines classical trajectories

Explicit solutions for simple symmetric symplectic spaces

Abstract

We present a definition of generating functions of canonical relations, which are real functions on symmetric symplectic spaces, discussing some conditions for the presence of caustics. We show how the actions compose by a neat geometrical formula and are connected to the hamiltonians via a geometrically simple variational principle which determines the classical trajectories, discussing the temporal evolution of such ``extended hamiltonians'' in terms of Hamilton-Jacobi-type equations. Simplest spaces are treated explicitly.