The Legendre transform, or transformation, seems to have many properties which are useful in different fields. For example:
- It switches between Lagrangian and Hamiltonian formalism in mechanics / tangent and cotangent bundle in differential geometry
- It is linked to variational problems (e.g. action)
- It is linked to convex analysis
- It is linked to statistical physics (e.g. free energy vs entropy)
- It is a sort of Fourier transform for the tropical semiring.
Where can I find all about it? Is there a reference specifically about the Legendre transform and all its properties?