On the gauge structure of the calculus of variations with constraints

TL;DR

This paper introduces a gauge-invariant geometric framework for constrained variational calculus, reformulating the problem using affine bundles and providing a new perspective on Pontryagin's maximum principle.

Contribution

It presents a novel gauge-invariant formulation of constrained variational calculus using affine scalar bundles and relates it to Pontryagin's maximum principle in a geometric setting.

Findings

Gauge-invariant formulation of constrained calculus

Equivalence between constrained and free variational problems in affine bundles

Geometric reinterpretation of Pontryagin's maximum principle

Abstract

A gauge-invariant formulation of constrained variational calculus, based on the introduction of the bundle of affine scalars over the configuration manifold, is presented. In the resulting setup, the Lagrangian is replaced by a section of a suitable principal fibre bundle over the velocity space. A geometric rephrasement of Pontryagin's maximum principle, showing the equivalence between a constrained variational problem in the state space and a canonically associated free one in a higher affine bundle, is proved.