Legendre transformation
In mathematics and physics, the Legendre transformation or Legendre transform, named after Adrien-Marie Legendre, is an involutive transformation on the real-valued convex functions of one real variable. Its generalization to convex functions of affine spaces is sometimes called the Legendre–Fenchel transformation.
It is commonly used in classical mechanics to derive the Hamiltonian formalism out of the Lagrangian formalism and in thermodynamics to derive the thermodynamic potentials, as well as in the solution of differential equations of several variables.
For sufficiently smooth functions on the real line, the Legendre transform f * of a function f can be specified, up to an additive constant, by the condition that the functions' first derivatives are inverse functions of each other,
Let I ⊂ ℝ be an interval, and f : I → ℝ a convex function; then its Legendre transform is the function f* : I* → ℝ defined by
where is the supremum, and the domain :
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