In mathematics and physics, the Legendre transformation, named after Adrien-Marie Legendre, is an involutive transformation on the real-valued convex functions of one real variable. 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.
The Legendre transform is an encoding of the convex hull of a function's epigraph in terms of it's supporting hyperplanes. If the function is convex and differentiable, then the supporting hyperplanes correspond to the derivative at each point, so the Legendre transform is a reencoding of a function's information in terms of it's derivative.
The Legendre Transform Ross Bannister, May 2005 Orthogonality of the Legendre polynomials The Legendre polynomials satisfy the following orthogonality property (1), d 1 x 1 xPn m 2 2n 1 mn 1 where is the th order Legendre polynomial. In meteorology it is sometimes convenient to integrate over the latitude domain,, instead of over.
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Making Sense of the Legendre Transform R. K. P. Zia1, Edward F. Redish2, and Susan R. McKay3 1Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061 USA 2Department of Physics, University of Maryland, College Park, MD 20742 USA and 3Department of Physics and Astronomy, University of Maine, Orono, ME.
The aim of this report is to list and explain the basic properties of the Legendre-Fenchel transform, which is a generalization of the Legendre transform commonly encountered in physics. The precise way in which the Legendre-Fenchel transform generalizes the Legendre transform is carefully explained and illustrated with many examples and pic-tures.
The Legendre transformation is a useful mathematical tool that is used in thermodynamics, classical mechanics and quantum field theory. Maybe the most famous application is that in classical mechanics, quantum mechanics and quantum field theory the Hamiltonian and the Lagrangian are connected by a Legendre transformation.
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Legendre transforms provide a means by which one can determine how the energy functions for different sets of thermodynamic variables are related. The general theory is given below for functions of a single variable.. where is known as the Legendre transform of. In shorthand notation, one writes however, it must be kept in mind that is a.