On Jacobi's condition for the simplest problem of calculus of variations with mixed boundary conditions

TL;DR

This paper extends Jacobi's criteria for the positive definiteness of the second variation to problems of calculus of variations with mixed boundary conditions, confirming the validity of Jacobi's condition in this broader context.

Contribution

It generalizes Jacobi's condition to include mixed boundary conditions in the simplest calculus of variations problems.

Findings

Jacobi's condition remains valid for mixed boundary conditions.

Extension applies to both constrained and isoperimetric problems.

Provides theoretical foundation for stability analysis with mixed boundaries.

Abstract

The purpose of this paper is the extension of Jacobi's criteria for positive definiteness of second variation of the simplest problems of calculus of variations subject to mixed boundary conditions. Both non constrained and isoperimetric problems are discussed. The main result is that Jacobi's condition remains valid also for the mixed boundary conditions.