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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?

Thanks!

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This might be a bit late but there is a really great expository article here and an even more basic article here. Both of these articles really helped me wrap my head around it.

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