Download PDF by Mariano Giaquinta, Stefan Hildebrandt (auth.): Calculus of Variations II

G on G given by f(x, z) = (x, cp(x, c», provided that r is ofform (10) and G =f(F).

In symplectic geometry the notion of a Lagrange manifold has been coined. This is an immersed N -dimensional submanifold of the 2N-dimensional space 1RN x 1RN annihilating the symplectic 2-form w = dYi 1\ dz i. In other words, a Lagrange manifold is an immersion u: 10 _1R N x 1RN of an N-dimensional parameter domain 10 such that u*w = Thus we obtain the following interpretation of Proposition 5. Suppose that u: 10 _1R N x 1RN are the initial values of a Hamiltonian flow h: 1 x 10 1R X 1RN X 1RN on a hyperplane {x = xo}, X o E 1, that is, o.

J. Fp , If (S. :1') is a solution of (17), we call Sz = Fp . ,Uj! a M ayer slape with the eikonal S. J. In terms of the Beltrami j()rm corresponding to F, (18) 30 Chapter 7. e. (20) and (17) means that ft*YF = dS. P = -%- and its inverse respectively. To this end we define the Cartan form K H on Q* by (22) Then we have (23) Let now f : r -+ G be a curve field in the configuration space IR x IRN with the slope &' and the slope field jt(x, z) = (x, z, &'(x, z». (x, z) = (x, z, lJI(x, and the dual slope function lJI(x, z) on G by z» (24) I/!

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