I have been studying the Lagrangian formulation of classical mechanics. I have been introduced to the Hamiltonian formulation via the Legendre transform, and studied the transform from this excellent introductory paper
Are the Lagrangian and Hamiltonian always Legendre transforms of each other? If so, does this apply to physics in general, or just to classical mechanics?
What happens if either of the Lagrangian or Hamiltonian is not convex? For example, if the former is not convex does this mean the Hamiltonian does not exist, and vice versa? The first answer to this question says "At the classical level, there is no convexity condition ...", but the meaning of this phrase is obscure and I feel the gist is really lost in the details. From my limited understanding of the Legendre transform it could only be applied to (strictly) convex functions.
- If either of the Lagrangian or Hamiltonian is not strictly convex this would mean non-unique values for the derivative that we need for the Legendre transformation. In this case, am I right in assuming that one would need to impose further conditions on the problem in order to restrict interest to a subset of the domain where the function in question is convex?