Let $$L(x,\dot x)=\frac{1}{2}m\dot x^2-\frac{1}{2}k(x-x_0)^2-mgx$$ the Lagrangian of a system. Euler Lagrange theorem says that a necessary condition to be a minimizer is to satisfy Euler-Lagrange condition. It's sufficient if the Lagrangian is convex. The method of Euler-Lagrange consist in finding solution of $$\frac{\partial L}{\partial x}=\frac{d}{dt}\frac{\partial L}{\partial \dot x},$$ and after, we say that the solution will give the minimizer. But since the Lagrangian is not convex, how can this method work?

  1. The underlying variational principle for EL equations is a principle of stationary action, despite that it is often referred to as a principle of least/minimal action.

  2. Also note that Lagrange equations may not have an action principle to begin with, cf. e.g. this Phys.SE post.

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  • $\begingroup$ Sorry, I don't really understand the meanning behind your answer. In this case we don't care about the convexity and the method works anyways ? If so, why ? $\endgroup$ – Dylan Feb 26 '19 at 22:05
  • $\begingroup$ I updated the answer. $\endgroup$ – Qmechanic Feb 26 '19 at 22:28

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