Symplectic Microgeometry II: Generating functions

TL;DR

This paper extends generating function techniques to symplectic microgeometry, demonstrating that symplectic micromorphisms always have global generating functions and applying this to Hamiltonian flows and classical mechanics.

Contribution

It introduces a categorical framework for symplectic micromorphisms with global generating functions, linking Hamilton-Jacobi solutions to symplectic microgeometry.

Findings

Every symplectic micromorphism admits a global generating function.

Hamiltonian flows can be described as special symplectic micromorphisms.

Provides a categorical formulation of classical mechanics evolution.

Abstract

We adapt the notion of generating functions for lagrangian submanifolds to symplectic microgeometry. We show that a symplectic micromorphism always admits a global generating function. As an application, we describe hamiltonian flows as special symplectic micromorphisms whose local generating functions are the solutions of Hamilton-Jacobi equations. We obtain a purely categorical formulation of the temporal evolution in classical mechanics.