论文标题
关于Tonelli变化的演讲的评论
A remark on Tonelli's calculus of variations
论文作者
论文摘要
本文提供了一种非常简单的方法，即具有积极的确定和超级线性拉格朗日的变体的计算方法。结果补充了Tonelli现代方法之前变体的经典文献。受Euler精神的启发，该提出的方法采用了确切的动作功能的有限维近似，其最小化可以轻松找到作为Euler对确切的Euler-Lagrange方程的离散化解决方案的解决方案。由近似最小化器生成的欧拉 - 库奇多边形线会收敛到精确的平滑最小化曲线。该框架在光滑曲线的家族中，在Lipschitz曲线家族中提供了一个基本的证明，并在平稳曲线的家族中的存在和规律性，因此，在没有对绝对连续的曲线和动作功能较低的半态度上使用现代功能分析。
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.
