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Actually the lagrangian is not minimized, it is merely stationary. An example of this is the harmonic oscillator, the lagrangian is clearly not bounded from below.



Yes, I realised that but I couldn't edit the post anymore :) I mean the action, not the Lagrangian, must have a minimum.


The action of a trajectory is not minimized either, it is usually at a saddle point, maximal relative to some trajectories and minimal to others. The linked paper has a detailed discussion of this.

http://www.eftaylor.com/pub/Gray&TaylorAJP.pdf




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