几何约束变分演算。I. - 分段光滑极值
Geometric Constrained Variational Calculus. I. - Piecewise smooth extremals
摘要：
提出了一种求解约束变分问题的几何方法。分析研究了在分段可微曲线上定义的作用泛函的极值，受可微的非完整约束。特别注意的是张量方面的理论。就运动学基础而言，通过引入无穷小控制的概念，发展了一个完全协变的方案。标准分类的极值正常和异常的讨论，指出存在一个代数算法分配给每个容许曲线相应的异常指数，相关的一个合适的线性映射的共同秩。注意力转移到研究的第一个变化的行动功能。分析包括一个修订庞特里亚金的方程和拉格朗日乘子法，以及重新制定庞特里亚金的算法在汉密尔顿条款。分析完成了一个一般的结果，关于存在的有限变形与固定端点。
英文摘要：
A geometric setup for constrained variational calculus is presented. The analysis deals with the study of the extremals of an action functional defined on piecewise differentiable curves, subject to differentiable, non-holonomic constraints. Special attention is paid to the tensorial aspects of the theory. As far as the kinematical foundations are concerned, a fully covariant scheme is developed through the introduction of the concept of infinitesimal control. The standard classification of the extremals into normal and abnormal ones is discussed, pointing out the existence of an algebraic algorithm assigning to each admissible curve a corresponding abnormality index, related to the co-rank of a suitable linear map. Attention is then shifted to the study of the first variation of the action functional. The analysis includes a revisitation of Pontryagin's equations and of the Lagrange multipliers method, as well as a reformulation of Pontryagin's algorithm in hamiltonian terms. The analysis is completed by a general result, concerning the existence of finite deformations with fixed endpoints.
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