For a function on a real vector space, its Legendre transformation is the convex conjugateWhen is differentiable and strictly convex, the maximizing point satisfies .
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The Legendre transformation is a mathematical operation used primarily in convex analysis and optimization, as well as in physics, particularly in thermodynamics and mechanics. It allows one to convert a function of one set of variables into a function of another set, changing the viewpoint on how the variables are related.
This is how you transform the Lagrangian into the Hamiltonian.