# The inverse problem of the calculus of variations for ordinary differential
equations

### About this Title

**Ian Anderson** and **Gerard Thompson**

Publication: Memoirs of the American Mathematical Society

Publication Year
1992: Volume 98, Number 473

ISBNs: 978-0-8218-2533-4 (print); 978-1-4704-0899-2 (online)

DOI: http://dx.doi.org/10.1090/memo/0473

MathSciNet review: 1115829

MSC: Primary 58E30; Secondary 49N45, 58H99

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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.

Readership

Research mathematicians in differential geometry and the
calculus of variations, exterior differential systems, and mathematical
physics.

### Table of Contents

**Chapters**

- 1. Introduction
- 2. The variational bicomplex for ordinary differential equations
- 3. First integrals and the inverse problem for second order ordinary
differential equations
- 4. The inverse problem for fourth order ordinary differential equations
- 5. Exterior differential systems and the inverse problem for second order
ordinary differential equations
- 6. Examples
- 7. The inverse problem for two dimensional sprays