In Lagrangian mechanics, we solve for a path for which, under variation, the first derivative is zero. This could be a minimum, maximum, or an inflection point. Any function which leads to true physical behavior can be considered to be a Lagrangian. Why do we not simply restrict the Lagrangian to one for which all valid true paths are local minimums? Are there some physical situations which defy such restrictions, or is it merely a mathematical coincidence?
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asked Nov 18, 2019 at 12:28
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For starters, the stationary path for an action is not at a minimum if there are conjugate points along the path. This is the generic situation and happens already for the harmonic oscillator, cf. e.g. my Phys.SE answer here.
answered Nov 18, 2019 at 12:44