The Geometry of Ordinary Variational Equations

Olga Krupková (auth.)3540638326, 9783540638322

The book provides a comprehensive theory of ODE which come as Euler-Lagrange equations from generally higher-order Lagrangians. Emphasis is laid on applying methods from differential geometry (fibered manifolds and their jet-prolongations) and global analysis (distributions and exterior differential systems). Lagrangian and Hamiltonian dynamics, Hamilton-Jacobi theory, etc., for any Lagrangian system of any order are presented. The key idea – to build up these theories as related with the class of equivalent Lagrangians – distinguishes this book from other texts on higher-order mechanics. The reader should be familiar with elements of differential geometry, global analysis and the calculus of variations.

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Table of contents :
Introduction….Pages 1-19
Basic geometric tools….Pages 20-40
Lagrangean dynamics on fibered manifolds….Pages 41-51
Variational Equations….Pages 52-79
Hamiltonian systems….Pages 80-96
Regular Lagrangean systems….Pages 97-128
Singular Lagrangean systems….Pages 129-148
Symmetries of Lagrangean systems….Pages 149-173
Geometric intergration methods….Pages 174-207
Lagrangean systems on π: R×M»R ….Pages 208-228