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Do solutions to the Euler Lagrange equation for physical Lagrangians actually minimize the action? In other words, is it known that for all Lagrangians used in application, that the unique solution to the Euler Lagrange equation subject to initial conditions actually is a global minimum for the action functional? How about local minimum? Are there known counterexamples?

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Solutions to the equations of motion stationarise the action, but do not need to be minima. For example, sphalerons are saddle points of the action of Electroweak theory.

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  • $\begingroup$ Are sphalerons physical? Are there examples of non-minima extremals (stationarizers in your parlance) that we have stronger experimental evidence for, in particular in the simpler domain of mechanics? I realize that Euler Lagrange only finds extremals, but my question is what comes up in practice, particularly in mechanics. Your example is interesting though; I'm not familiar enough with QFT to make much sense of it though. $\endgroup$ Jun 10, 2020 at 21:51
  • $\begingroup$ Sphalerons are physical insofar as that they are widely accepted to be solutions of the standard model equations, though they occur at too high energies to have been observed (a bit like Hawking radiation is widely accepted as physical but hasn’t been observed) $\endgroup$
    – DavidH
    Jun 11, 2020 at 7:15
  • $\begingroup$ In general insantons are solutions of the equations of motion of a system that are often not minima, and are relevant for quantum tunnelling calculations. You can do predictive quantum tunnelling calculations using instantons $\endgroup$
    – DavidH
    Jun 11, 2020 at 7:17

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