A Remark on Tonelli’s Calculus of Variations

    Received 13 December 2022; accepted 31 March 2023; published 12 May 2023

    2023, Vol. 19, no. 2, pp.  239-248

    Author(s): Soga K.

    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
    Citation: Soga K., A Remark on Tonelli’s Calculus of Variations, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 2, pp.  239-248

    Download File
    PDF, 286.03 Kb

    Creative Commons License
    This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License