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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 ...
Legendre wavelet. In functional analysis, compactly supported wavelets derived from Legendre polynomials are termed Legendre wavelets or spherical harmonic wavelets. [ 1] Legendre functions have widespread applications in which spherical coordinate system is appropriate. [ 2][ 3][ 4] As with many wavelets there is no nice analytical formula for ...
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).
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 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.
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 .
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.
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given (at least locally) as the kernel of a differential one-form, and the non-integrability ...