The Geometry of Ordinary Variational Equations

1. Front Matter

   Pages I-X

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2. Introduction

   • Olga Krupková

   Pages 1-19

3. Basic geometric tools

   • Olga Krupková

   Pages 20-40

4. Lagrangean dynamics on fibered manifolds

   • Olga Krupková

   Pages 41-51

5. Variational Equations

   • Olga Krupková

   Pages 52-79

6. Hamiltonian systems

   • Olga Krupková

   Pages 80-96

7. Regular Lagrangean systems

   • Olga Krupková

   Pages 97-128

8. Singular Lagrangean systems

   • Olga Krupková

   Pages 129-148

9. Symmetries of Lagrangean systems

   • Olga Krupková

   Pages 149-173

10. Geometric intergration methods

    • Olga Krupková

    Pages 174-207

11. Lagrangean systems on π: R×M»R

    • Olga Krupková

    Pages 208-228

12. Back Matter

    Pages 229-251

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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.

• Hamilton-Jacobi theory
• Hamiltonian mechanics
• Lagrangian mechanics
• calculus
• differential equation
• differential geometry
• higher-order variational equations
• manifold
• regularity

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