Let $$ r'' = \mathrm{F}(r', r)$$

be Newton equation in one variable whith $\mathrm{F}$ locally Lipschitz.

Is there a function $\mathcal{L}(r',r)$ such that the Newton equation is in fact Euler-Lagrange equation $$ \frac{\partial \mathcal{L}}{\partial r} = \frac{d}{dt}\frac{\partial \mathcal{L}}{\partial r'} \quad ?$$

I know that it is true for a conservative force field, but what about the general case?


your first question is also known as the inverse problem for the Lagrangian mechanics.

Answering your question, there exists a necessary and sufficient condition in order that a second order ode $F(r'',r',r)=0$ be of Euler-Lagrange kind. This condition, called after Helmholtz, is expressed by the system $$\left\{\begin{gathered} \frac{\partial F_i}{\partial r_j''}=\frac{\partial F_j}{\partial r_i''},\\ \frac{\partial F_i}{\partial r_j'}+\frac{\partial F_j}{\partial r_i'}=2\frac{d}{dt}\frac{\partial F_j}{\partial r_i''},\\ \frac{\partial F_i}{\partial r_j}-\frac{\partial F_j}{\partial r_i}=\frac{1}{2}\frac{d}{dt}\left(\frac{\partial F_i}{\partial r_j'}-\frac{\partial F_j}{\partial r_i'}\right), \end{gathered}\right.$$ where the Einstein summation convention is active.


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