Legendre Transformation

The Legendre transformation is an involutive transformation on real valued functions that are convex in a real variable, first introduced by Adrien-Marie Legendre in 1787. It converts a function of one quantity into a function of a conjugate quantity, for example turning a position dependent function into a momentum dependent one, by exchanging the original variable for the value of the functions own derivative. For a differentiable convex function, the transform can be characterized by the property that the derivatives of the original and transformed functions are inverse functions of each other, and it can be pictured geometrically as encoding a curve by the slopes and intercepts of its tangent lines rather than by its points. The transformation is essential in classical mechanics for converting between the Lagrangian and Hamiltonian formulations, in thermodynamics for deriving potentials such as enthalpy and Gibbs free energy, and in probability theory for computing rate functions in large deviations theory. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/