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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.
Received: 13.12.2023 Accepted: 31.03.2023
Citation:
K. Soga, “A Remark on Tonelli’s Calculus of Variations”, Rus. J. Nonlin. Dyn., 19:2 (2023), 239–248
Linking options:
https://www.mathnet.ru/eng/nd850 https://www.mathnet.ru/eng/nd/v19/i2/p239
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Abstract page: | 42 | Full-text PDF : | 12 | References: | 8 |
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