Legendre transform

A mathematical transformation that replaces selected variables of a function with their conjugate derivatives. It encodes the same information in a different set of variables when the function is suitably convex or concave.

Convex conjugate: The supremum of affine functions formed from a function and a slope variable; also called the Fenchel conjugate. It generalizes the Legendre transform and clarifies its definition for nonsmooth functions.
Hamiltonian mechanics: A formulation of classical mechanics using generalized positions and momenta to describe the evolution of a system. The Hamiltonian is obtained from the Lagrangian by transforming velocities into conjugate momenta.
Derivative: A measure of a function's instantaneous rate of change, defined through a limiting ratio of increments. The transformed variable is the derivative with respect to the variable being replaced.
Fourier transform: An integral transform that represents a function in terms of its frequency components. It changes representation through oscillatory integration, rather than through convex slope optimization.
Convex function: A function whose value along every line segment is no greater than the corresponding linear interpolation of endpoint values. Convexity ensures the slope identifies a supporting tangent and helps recover the original function.
Thermodynamic potentials: State functions that encode thermodynamic energy under different choices of natural variables. Legendre transforms generate potentials such as enthalpy and Gibbs free energy from internal energy.
Tangent line: A line that locally matches a curve's direction at a point. For a differentiable convex function, the derivative sets the slope of a supporting tangent.
Laplace transform: An integral transform that maps a time-domain function to a complex-frequency representation. It also changes variables, but uses an exponential kernel instead of conjugate derivatives.
Young's inequality: An inequality relating a convex function to its convex conjugate, with equality when their arguments are paired by a derivative. It expresses the inequality underlying the conjugate construction and its equality condition.
Lagrangian mechanics: A formulation of classical mechanics in which motion follows from a function of generalized positions and velocities. Its velocity variables are exchanged for momenta in the transition to Hamiltonian mechanics.

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