Elimination of Hamilton-Jacobi equation in extreme variational problems

TL;DR

This paper demonstrates that certain one-dimensional Euler-Lagrange variational problems can be solved without the Hamilton-Jacobi equation, simplifying the process under specific conditions and extending results to compact extrema.

Contribution

It introduces a method to solve extremal problems without Hamilton-Jacobi equations under strengthened Legendre conditions, with extensions to compact extrema in Sobolev spaces.

Findings

Solution without Hamilton-Jacobi equation under Legendre condition

Restriction on interval length based on integrand form

Extension to compact extrema in H^1[a;b]

Abstract

It is shown that extreme problem for one-dimensional Euler-Lagrange variational functional in $C^1[a;b]$ under the strengthened Legendre condition can be solved without using Hamilton-Jacobi equation. In this case, exactly one of the two possible cases requires a restriction to a length of $[a;b]$, defined only by the form of integrand. The result is extended to the case of compact extremum in $H^1[a;b]$.