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Additional info for Calculus of Variations II. The Hamilton Formalism: The Hamiltonian Formalism: v. 2
Hence fxx > 0 implies ff*4 > 0, and vice versa. In other words, the Legendre transform f * of a uniformly convex (concave) function f : 0 -+ 1R is again a uniformly convex (concave) function provided that 0* := f(Q) is convex. The function f * : 92* -+ lR is sometimes called the conjugate convex (concave) function to f. Here a function f :0 Ht is called uniformly convex (concave) if 12 is a convex open set and if it is a C'-function satisfying fsx > 0 (fXZ < 0). 0and x#z. I 1 Gradient Mappings and Legendre Transformations 9 IY Fig.
54=0, (c2 - u2)ux - uv(u, + vx) + (C2 - v2)vY = 0, where c is the speed of sound which is a given function of u2 + v2. ' + (C2-f2)f=0. Then the Legendre transform f *(, n) solves the linear second order differential equation (C2 2)f + 2 nfCn + (C2 n2)fCe = 0. f* = A(C, n). 3 Let A = (aae) be a symmetric invertible matrix with the inverse A-' = (00), and consider the nondegenerate quadratic form f(x) ='-zaa,xax6 Note that f(x) is not necessarily convex as A is merely invertible and can be nondefinite Its gradient mapping is given by C = f,(x) = Ax or s = a,yxo, whence x = f4*(C) = A-' or x° = a'% and the Legendre transform f * of f is f*(C) = za"CaCB.
If (GA) holds, the tensor T can be pushed forward onto the domain G* = 2(G) by setting H := T o 2'. (x, z, q$,,(x, z, iv)). Taking (9) into account, we obtain (12) Ha = [0 - °ong)ba + 7rPo*-. If n = 1, the tensor (HQ) has the only component Hi = ¢, and therefore H can be identified with the Hamilton function 0. For the sake of simplicity we again denote H = (HQ(x, z, n)) as Hamilton tensor. In the calculus of variations, the tensors T and H were apparently for the first time used by Caratheodory while they appeared much earlier in physics, for instance in Maxwell's theory of 1 2 Legendre Duality Between Phase and Cophase Space 21 electromagnetism and in relativity theory.