A remark on Tonelli's calculus of variations

TL;DR

This paper introduces a simple, elementary method for the calculus of variations with positive definite, superlinear Lagrangians, using finite dimensional approximation to establish existence and regularity of minimizers without advanced functional analysis.

Contribution

It offers a novel, elementary approach to proving the existence and regularity of minimizers in calculus of variations, complementing classical methods with finite dimensional approximation.

Findings

Elementary proof of existence of smooth minimizers

Convergence of Euler-Cauchy polygonal lines to smooth curves

Extension to Lipschitz minimizers with minimal additional steps

Abstract

This paper provides a quite simple method of Tonelli's calculus of variations with positive definite and superlinear Lagrangians. The result complements the classical literature of calculus of variations before Tonelli's modern approach. Inspired by Euler's spirit, the proposed method employs finite dimensional approximation of the exact action functional, whose minimizer is easily found as a solution of Euler's discretization of the exact Euler-Lagrange equation. The Euler-Cauchy polygonal line generated by the approximate minimizer converges to an exact smooth minimizing curve. This framework yields an elementary proof of the existence and regularity of minimizers within the family of smooth curves and hence, with a minor additional step, within the family of Lipschitz curves, without using modern functional analysis on absolutely continuous curves and lower semicontinuity of action functionals.