2015/03/16 by Enrico Massa, Danilo Bruno, Gianvittorio Luria +1
Engineering · Mathematics · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #Algebra over a field #Applied mathematics #Calculus (dental) #Calculus of variations #Elasticity and Material Modeling #Mathematical analysis #Mathematical optimization #Mathematics #Numerical methods for differential equations #Piecewise #Pure mathematics #math-ph #math.MP #msc:37J #msc:49J #msc:70F25
paper · pdf · doi:10.1142/s0219887815500619
published as Int. J. Geom. Methods Mod. Phys. Vol. 12 (2015), 1550061 · 30 pages. arXiv admin note: substantial text overlap with arXiv:0705.2362
openalex publication_date 2015/03/16 · arxiv created 2015/03/30 · arxiv updated 2015/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.