Hamiltonian Structure of Gauge-Invariant Variational Problems

TL;DR

This paper investigates the geometric structure of solutions to gauge-invariant variational problems, revealing an affine fiber-bundle structure over the Euler-Lagrange solutions and exploring related spaces like Jacobi fields and moduli spaces.

Contribution

It establishes the affine fiber-bundle structure of Hamilton-Cartan solutions over Euler-Lagrange solutions for gauge-invariant Lagrangians, extending understanding of the solution space geometry.

Findings

Solutions form an affine fiber-bundle over Euler-Lagrange solutions.

Structure of Jacobi fields is analyzed within this framework.

Moduli space of extremals is also studied.

Abstract

Let $C\to M$ be the bundle of connections of a principal bundle on $M$. The solutions to Hamilton-Cartan equations for a gauge-invariant Lagrangian density $\Lambda $ on $C$ satisfying a weak condition of regularity, are shown to admit an affine fibre-bundle structure over the set of solutions to Euler-Lagrange equations for $\Lambda $. This structure is also studied for the Jacobi fields and for the moduli space of extremals.