Geodesic
The principle of stationary action in the calculus of variations
Communications in Mathematics, Tome 20 (2012) no. 2, pp. 89-116.

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We review some techniques from non-linear analysis in order to investigate critical paths for the action functional in the calculus of variations applied to physics. Our main intention in this regard is to expose precise mathematical conditions for critical paths to be minimum solutions in a variety of situations of interest in Physics. Our claim is that, with a few elementary techniques, a systematic analysis (including the domain for which critical points are genuine minima) of non-trivial models is possible. We present specific models arising in modern physical theories in order to make clear the ideas here exposed.
Classification : 34K11, 49K15, 49S05
Keywords: stationary action; functional extrema; conjugate points; oscillatory solutions; Lane-Emden equations 
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     title = {The principle of stationary action in the calculus of variations},
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López, Emanuel; Molgado, Alberto; Vallejo, José A. The principle of stationary action in the calculus of variations. Communications in Mathematics, Tome 20 (2012) no. 2, pp. 89-116. http://geodesic.mathdoc.fr/item/COMIM_2012__20_2_a3/