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The function () is defined on the interval [,].For a given , the difference () takes the maximum at ′.Thus, the Legendre transformation of () is () = ′ (′).. In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, [1] is an involutive transformation on real-valued functions that are ...
In mathematics, Legendre transform is an integral transform named after the mathematician Adrien-Marie Legendre, which uses Legendre polynomials as kernels of the transform. Legendre transform is a special case of Jacobi transform. The Legendre transform of a function () is [1] [2] [3] {()} = ~ = The inverse Legendre transform is given by
In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation, or Fenchel conjugate (after Adrien-Marie Legendre and Werner Fenchel).
Inverse two-sided Laplace transform; Laplace–Carson transform; Laplace–Stieltjes transform; Legendre transform; Linear canonical transform; Mellin transform. Inverse Mellin transform; Poisson–Mellin–Newton cycle; N-transform; Radon transform; Stieltjes transformation; Sumudu transform; Wavelet transform (integral) Weierstrass transform ...
The finite Legendre transform (fLT) transforms a mathematical function defined on the finite interval into its Legendre spectrum. [1] [2] Conversely, the inverse fLT (ifLT) reconstructs the original function from the components of the Legendre spectrum and the Legendre polynomials, which are orthogonal on the interval [−1,1].
Legendre transformation, which converts between the Lagrangian and Hamiltonian, is an involutive operation. Integrability, a central notion of physics and in particular the subfield of integrable systems, is closely related to involution, for example in context of Kramers–Wannier duality.
The relation between the two is by a Legendre transformation, and the condition that determines the classical equations of motion (the Euler–Lagrange equations) is that the action has an extremum. In quantum mechanics, the Legendre transform is hard to interpret, because the motion is not over a definite trajectory.
The left hand side is the Legendre transformation of with respect to ′ (). The intuition behind this result is that, if the variable x {\displaystyle x} is actually time, then the statement ∂ L ∂ x = 0 {\displaystyle {\frac {\partial L}{\partial x}}=0} implies that the Lagrangian is time-independent.