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Given an action, I can get the differential equation governing the evolution of the system by applying the principle of least action.

Does it work the other way around? Given any differential equation is there an "action" which can be varied to obtain it?

EDIT: Including GR for example. A set of 10 coupled nonlinear differential equations.

dan-ros
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1 Answers1

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All self-adjoint differential equations are consistent with the Principle of Least Action. See p. 226 of Lanczos' Linear Differential Operators.

Lanczos explains that those physical systems which exhibit no loss of energy automatically provide a scalar quantity which can be minimized/maximized.

For more advanced cases see How do I show that there exists variational/action principle for a given classical system?

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Peter Diehr
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