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.


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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  • $\begingroup$ How does this argument mesh with general relativity where only in situations where we have a time like killing vector is energy conserved? $\endgroup$ – Mason Apr 5 '16 at 16:54
  • $\begingroup$ @Mason: I'm not a GR expert, but the usual process is to start with the Ricci scalar curvature, and I found this historical exposition of Hilbert's derivation.. I can't speak to the energy conservation in detail. $\endgroup$ – Peter Diehr Apr 5 '16 at 18:51
  • $\begingroup$ I know how the Hilbert action is derived, and I know one need make no reference to energy. It's derived by assuming that the equations of motion are second order, generally covariant and satisfy the equivalence principal. I just find it hard to understand how Lanczos' argument based on systems conserving energy explains how you have an action in GR. In general relativity, there is no time evolution and hence no generator of such time evolutions, so there is no such thing as energy built into the system. $\endgroup$ – Mason Apr 6 '16 at 0:35
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    $\begingroup$ Lanczos does not claim that all systems obeying least action principals conserve energy, he instead claims that the systems which conserve energy obey least action principals. I think that your mention of energy dissipation is misleading and should be edited out since there is at least one very well known system with no energy which does have an associated scalar action which can be varied to give it's equations of motion. $\endgroup$ – Mason Apr 6 '16 at 1:18
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    $\begingroup$ Lanczos only talks about linear eoms. The full problem of whether or not a given set of eoms can be deduced (possible via a transformation) from a variational principle is typically an open problem, cf. my Phys.SE answer here. $\endgroup$ – Qmechanic Apr 6 '16 at 12:18

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