Mathematical problems of nonlinearity

A Remark on Tonelli’s Calculus of Variations

K. Soga

Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522 Japan

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.

Keywords: Tonelli’s calculus of variations, direct method, action minimizing, minimizing curve, regularity of minimizer, Euler method, Euler – Cauchy polygon.

MSC: 49J15, 49M25, 37J51

Received: 13.12.2023
Accepted: 31.03.2023

Language: English

DOI: 10.20537/nd230501