The Legendre transform - University of Toronto.

The main application of and the historical root of the notion of Legendre transform (in differential geometry) is in classical physics and its formalization by symplectic geometry. In classical mechanics, the Hamiltonian function H H is a Legendre transform of the Lagrangian L L and vice versa.

Topic 2070 Legendre Transformations Many important thermodynamic equations are closely related. These relationships are often highlighted by the mathematical technique, Legendre Transformations (1,2). With reference to thermodynamics, Callen (3) discusses application of Legendre Transformations.

Legendre-Fenchel is one such transform which maps the (x, f(x)) space to the space of slope and con- jugate that is (p, f(p)). However, while the Fourier transform consists of an integration with a kernel, the Legendre Fenchel transform uses supremum as the transformation procedure.

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.

Dirac Structures and the Legendre Transformation for Implicit Lagrangian and Hamiltonian Systems Hiroaki Yoshimura Mechanical Engineering, Waseda University Tokyo, Japan Joint Work with Jerrold E. Marsden Control and Dynamical Systems, Caltech.

When this professor had a hard time having students understand Legendre transformation, he converted the act of learning the Legendre transformation into a kind of laboratory exercise in which the students were forced to grapple with the problem in his or her own way.

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