Geometric Constrained Variational Calculus. I. - Piecewise smooth extremals

TL;DR

This paper develops a geometric framework for constrained variational calculus on piecewise smooth curves, introducing covariant schemes, tensorial analysis, and algebraic algorithms for classifying extremals, including abnormal ones.

Contribution

It introduces a fully covariant geometric setup for constrained variational calculus, including a new algebraic method to classify extremals and analyze their properties.

Findings

Developed a covariant scheme using infinitesimal control.

Created an algebraic algorithm for abnormality index classification.

Reformulated Pontryagin's equations and Hamiltonian methods.

Abstract

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.