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

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

by Ian Anderson, Gerard Thompson
     
 

ISBN-10: 082182533X

ISBN-13: 9780821825334

Pub. Date: 07/14/1992

Publisher: American Mathematical Society

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

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Overview

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.

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Product Details

ISBN-13:
9780821825334
Publisher:
American Mathematical Society
Publication date:
07/14/1992
Series:
Memoirs Series, #47
Pages:
110
Product dimensions:
7.09(w) x 10.04(h) x (d)

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