Constrained variational calculus: the second variation (part I)

TL;DR

This paper develops a covariant framework for analyzing the second variation in constrained calculus of variations, providing conditions for minimality and linking them to Jacobi fields within a geometric setting.

Contribution

It introduces a fully covariant representation of the second variation for constrained variational problems, including necessary and sufficient minimality conditions, and relates them to Jacobi fields.

Findings

Explicit covariant second variation formula derived

Necessary and sufficient minimality conditions established

Connection to Jacobi fields clarified

Abstract

Within the geometrical framework developed in arXiv:0705.2362, the problem of minimality for constrained calculus of variations is analysed among the class of differentiable curves. A fully covariant representation of the second variation of the action functional, based on a suitable gauge transformation of the Lagrangian, is explicitly worked out. Both necessary and sufficient conditions for minimality are proved, and are then reinterpreted in terms of Jacobi fields.