By Mariano Giaquinta, Stefan Hildebrandt
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Additional info for Calculus of Variations II. The Hamilton Formalism: The Hamiltonian Formalism: v. 2
4 Chapter 7. 2. In Section 3 we shall give an exposition of the notions of a convex body and its polar body as well as of a convex function and its conjugate. This way we are led to a generalized Legendre transformation which will be used in Chapter 8 to develop a canonical formalism for one-dimensional parametric variational problems. The last subsection explores some ramifications of the theory of convex functions which are of use in optimization theory and for the direct methods of the calculus of variations based on the notion of lower semicontinuity of functionals.
The integrability conditions (16) take the simple form (30) 8Y% a - -8H aZi' aVIi - 0Y/k 05k - azI ' where H(x, z) := H(x, z, W(x, z)), and the Caratheodory equations (17) are just (31) SX = -H(x, z, YP), SS = V. 1. Canonical Equations and the Partial Differential Equation of Hamilton-Jacobi 31 These equations imply the Hamilton-Jacobi equation Sx + H(x, z, Sz) = 0. (32) Thus we have found that the eikonal S(x, z) of an arbitrary Mayer field f on G satisfies (32). Let conversely SC C2(G) be a solution of (32).
Legendre Transformation, Hamiltonian Systems, Convexity, Field Theories yj = F, (x, z, p), P' = H,, (x, z, y), F(x, z, p) + H(x, z, y) = yip', Fx(x, z, p) + Hx(x, z, y) = 0, FZ,(x, z, p) + HZ,(x, z, y) = 0 if (x, z, p) _ 9-1(x, z, y) or (x, z, y) = 2(x, z, p). e. the Legendre transformation (1), (3) is involutory. Consider now an F-extremal u e CZ([a, b], RN) whose 1-graph is contained in Q, and set n(x) := u'(x). The the "prolongation" e(x) := (x, u(x), it(x)) of u(x) satisfies the Euler equations d du (5) dx = 7r, dxF(e) = FZ(e).
Calculus of Variations II. The Hamilton Formalism: The Hamiltonian Formalism: v. 2 by Mariano Giaquinta, Stefan Hildebrandt