# The Inverse Problem of the Calculus of Variations for Ordinary Differential Equations

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*Ian Anderson; Gerard Thompson*

This monograph explores various aspects of the inverse problem of the calculus of variations for systems of ordinary differential equations. The main problem centers on determining the existence and degree of generality of Lagrangians whose system of Euler-Lagrange equations coincides with a given system of ordinary differential equations. The authors rederive the basic necessary and sufficient conditions of Douglas for second order equations and extend them to equations of higher order using methods of the variational bicomplex of Tulcyjew, Vinogradov, and Tsujishita. What emerges is a fundamental dichotomy between second and higher order systems: the most general Lagrangian for any higher order system can depend only upon finitely many constants. The authors present an algorithm, based upon exterior differential systems techniques, for solving the inverse problem for second order equations. A number of new examples illustrate the effectiveness of this approach. The monograph also contains a study of the inverse problem for a pair of geodesic equations arising from a two dimensional symmetric affine connection. The various possible solutions to the inverse problem for these equations are distinguished by geometric properties of the Ricci tensor.

#### Table of Contents

# Table of Contents

## The Inverse Problem of the Calculus of Variations for Ordinary Differential Equations

- Table of Contents v6 free
- Abstract vi7 free
- 1. Introduction 18 free
- 2. The Variational Bicomplex for Ordinary Differential Equations 1219 free
- 3. First Integrals and the Inverse Problem for Second Order Ordinary Differential Equations 2734
- 4. The Inverse Problem for Fourth Order Ordinary Differential Equations 3542
- 5. Exterior Differential Systems and the Inverse Problem for Second Order Ordinary Differential Equations 5057
- 6. Examples 6471
- 7. The Inverse Problem for Two Dimensional Sprays 8895
- 8. References 108115