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arXiv · 0710.4532

On the action principle for a system of differential equations

D. M. Gitman·V. G. Kupriyanov
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Abstract

We consider the problem of constructing an action functional for physical systems whose classical equations of motion cannot be directly identified with Euler-Lagrange equations for an action principle. Two ways of action principle construction are presented. From simple consideration, we derive necessary and sufficient conditions for the existence of a multiplier matrix which can endow a prescribed set of second-order differential equations with the structure of Euler-Lagrange equations. An explicit form of the action is constructed in case if such a multiplier exists. If a given set of differential equations cannot be derived from an action principle, one can reformulate such a set in an equivalent first-order form which can always be treated as the Euler-Lagrange equations of a certain action. We construct such an action explicitly. There exists an ambiguity (not reduced to a total time derivative) in associating a Lagrange function with a given set of equations. We present a complete description of this ambiguity. The general procedure is illustrated by several examples.

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D. M. Gitman, V. G. Kupriyanov. 2007-10-31. On the action principle for a system of differential equations. https://doi.org/10.1088/1751-8113/40/33/010

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