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Is it possible to model any physical system with a Lagrangian convex in its velocity variables ?

I am aware that many Lagrangian can model the same system and maybe not all of them are partially convex in their velocity variables, but can we always find one ?

Do we have an understanding about what kinds of systems cannot be described by such a Lagrangian ?

Most of the Lagrangians I worked with have this property which is useful to derive a Hamiltonian through the Legendre transform and I was wondering to which extent this property holds and how fundamental it is ? Do "most" physical systems have it ?

I am aware of similar questions on this forum, I went through them but I don't believe any of the posts really answers my question.

Qurious Spirit
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  • This question is a bit vague owing to the word "any". Certainty you ordinarily expect velocities to contribute as a square ($v^2$), and standard defn of kinetic energy agrees with that. But are you intending to construct models for toy systems in order to explore ideas? Are you intending to do quantum field theory? What have you got in mind? – Andrew Steane May 09 '20 at 10:39
  • I'm just trying to understand if I lose anything or what do I lose if by searching for a physical theory, I restrict myself to Lagrangians that are convex in velocity terms ? Are there any physical phenomea that cannot be captured by partially convex Lagrangians ? Not aiming for QFT for now. Understanding the limitations of such an approach (modeling with convex Lagrangians in velocities) for non quantum systems would be very interesting to me. – Qurious Spirit May 10 '20 at 03:52
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    Does this answer your question? – J. Murray May 27 '20 at 05:36
  • Not really, the counter example (convex Lagrangian but not strictly convex) answers their question but not mine. – Qurious Spirit Jun 08 '20 at 03:25

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