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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 convex on a real variable. Specifically, if a real-valued multivariable function is convex on one of its ...
Convex conjugate. 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).
The Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre [3] as the coefficients in the expansion of the Newtonian potential where r and r′ are the lengths of the vectors x and x′ respectively and γ is the angle between those two vectors. The series converges when r > r′.
The Lagrangian is thus a function on the jet bundle J over E; taking the fiberwise Legendre transform of the Lagrangian produces a function on the dual bundle over time whose fiber at t is the cotangent space T ∗ E t, which comes equipped with a natural symplectic form, and this latter function is the Hamiltonian.
Legendre form. In mathematics, the Legendre forms of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced. Legendre chose the name elliptic integrals because [ 1] the second kind gives the arc length of an ellipse of unit semi-major axis and eccentricity (the ellipse being defined parametrically ...
Legendre transform (integral transform) 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 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 Legendre transform of () = is () = + , hence + for all non-negative and . This estimate is useful in large deviations theory under exponential moment conditions, because b ln b {\displaystyle b\ln b} appears in the definition of relative entropy , which is the rate function in Sanov's theorem .