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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 of a function can be specified, up to an additive constant, by the condition that the functions' first derivatives are inverse functions of each other, : ==Definition== Let be an interval, and a convex function; then its ''Legendre transform'' is the function defined by : with domain : The transform is always well-defined when is convex. The generalization to convex functions on a convex set is straightforward: has domain : and is defined by : where denotes the dot product of and . The function is called the convex conjugate function of . For historical reasons (rooted in analytic mechanics), the conjugate variable is often denoted , instead of . If the convex function is defined on the whole line and is everywhere differentiable, then : can be interpreted as the negative of the of the tangent line to the graph of that has slope . The Legendre transformation is an application of the duality relationship between points and lines. The functional relationship specified by can be represented equally well as a set of points, or as a set of tangent lines specified by their slope and intercept values. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Legendre transformation」の詳細全文を読む スポンサード リンク
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