The geometry of ordinary variational equations
type: Item
instance of: scholarly article
title: The geometry of ordinary variational equations
zbMATH Open document ID: 0936.70001
author: Olga Rossi
published in: Lecture Notes in Mathematics
publication date: 1997-12-17T00:00:00Z
review text: As stated in the preface, the goal of this work is ``to provide a general, comprehensive and self-contained geometric theory of ordinary differential equations which are variational''. The presented intrinsic framework from differential geometry and global analysis allows to handle also nonautonomous and degenerate situations in a unified fashion. The introduction provides motivation for the rest of the text in the form of a review of historical problems and their solutions, starting from the standard derivation of the Euler-Lagrange equations in coordinates, Bernoulli's brachystochrone problem, the inverse problem of the calculus of variations (find the conditions on the coefficients of a second order ordinary differential equation such that the later is variational, i.e. takes the form of an Euler-Lagrange equation) stated by Helmholtz and endowed with a recent proof, the notion of equivalent Lagrangians (i.e. their Euler-Lagrange equations coincide), conditions on a connection so that its geodesics are Euler-Lagrange equations, the form of the potential so that the equations describing a mechanical holonomic system are variational (one sees as a consequence that the origin of the form of the Lorentz force as well as the Coriolis effect is variational), and the derivation of Hamilton's equations in the regular case under a Legendre transformation for integration. Chapter two provides the language and the notations so that the rest of the text can be read. One starts with distributions and their symmetries. Frobenius theorem, and one continues with presymplectic forms, their representations in coordinates via Darboux theorem, and with their symmetries. One defines next the fundamental object, i.e. a jet prolongation of a fibred manifold constructed from its jets of sections. The base space is, in general, one-dimensional and means physically time (this is why the framework is well suited to nonautonomous Lagrangians). In this case coordinate representations are pre…
Mathematics Subject Classification ID: 70-02
Mathematics Subject Classification ID: 70H50
Mathematics Subject Classification ID: 49K15
Mathematics Subject Classification ID: 58E30
Mathematics Subject Classification ID: 70H03
Mathematics Subject Classification ID: 70H20
Mathematics Subject Classification ID: 70H33
zbMATH DE Number: 1097467
zbMATH Keywords: Lepagean-form
zbMATH Keywords: inverse problem of calculus of variations
zbMATH Keywords: Jacobi complete integral
zbMATH Keywords: Euler-Lagrange equations
zbMATH Keywords: Legendre transformation
zbMATH Keywords: jet prolongation
zbMATH Keywords: integral sections
zbMATH Keywords: Lagrangian dynamics
zbMATH Keywords: fibre manifolds
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: Hamilton extremals
zbMATH Keywords: Euler-Lagrange distribution
zbMATH Keywords: singular Lagrangian systems
zbMATH Keywords: holonomic sections
zbMATH Keywords: symmetries
zbMATH Keywords: Hamilton-Jacobi equation
zbMATH Keywords: semiregular Lagrangian systems
zbMATH Keywords: quasi-configuration space
zbMATH Keywords: Hamilton-Jacobi distribution
reviewed by: A. Akutowicz
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0093438
OpenAlex ID: W1564260361
DOI: 10.1007/BFB0093438
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description: scientific article; zbMATH DE number 1097467