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I am interested in the following matter.

Given an ODE, can one always find a functional F such that the ODE is its Euler Lagrange equation?

I am thinking at the following concrete case.

I have the ODE y' = a y

I would like a functional given by the intergral over a line of a function F = F (y, y'), such that the E.L. equation, giving necessary conditions for stationarity of the functional, is my ODE.

Any help is really welcome

Thanks

Muzialis