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All of the toy problems in Lagrangian mechanics I have come across are analytic. Most of the non-analytic functions I know don't seem to appear in Lagrangian mechanics. I can, of course, see how a carefully constructed constraint would trivially make the Lagrangian non-analytic, but the region of smooth-but-not-analytic functions seems to be filled with abstract functions that don't seem amenable to a natural physical construction.

Do we come across smooth but non-analytic Lagrangians in practical unconstrainted scenarios, or do they only show up in systems with intentionally constructed constraints shaped to have this property?

For context, the property I am looking to explore is the ability to use Taylor polynomials to approximate a Lagrangian from just a single point, using only higher order derivatives to do so.

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Well, for what it's worth, in the realm of effective field theories (EFT), say under a RG flow, one may ponder what class of functions the Lagrangian density should belong to, e.g. analytic, $C^{\infty}$-smooth, or something third, in order to e.g. ensure that the corresponding Euler-Lagrange (EL) equations have solutions.

In this context it seems relevant to mention Lewy's & Mizohata's examples [1] of linear PDEs with no solutions for some non-analytic $C^{\infty}$-smooth source functions.

References:

  1. P.J. Olver, Applications of Lie Groups to Differential Equations, 1993; p. 160.
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