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Unformatted text preview: 1 Homework #10 1. Consider the functional with Lagrangian F ( x,u,p ) = p x 2 + u 2 p 1 + p 2 . (a) Find the Hamiltonian and the associated Hamiltonian system. Solution Find that π = ∂F ∂p = p √ x 2 + u 2 √ 1+ p 2 , so that (after simplifying), the Hamiltonian is H ( x,u,π ) = πp- F ( x,u,π ) =- p x 2 + u 2- π 2 . The Hamiltonian system is du dx = ∂H ∂π = π √ x 2 + u 2- π 2 , dπ dx =- ∂H ∂u = u √ x 2 + u 2- π 2 . (b) Show that u 2- π 2 is a conserved quantity. Solution Using the chain rule, d dx ( u 2- π 2 ) = 2 u du dx- 2 π dπ dx . From part (a), this derivative is zero, so u 2- π 2 is indeed conserved. (c) Find a solution to the Hamilton-Jacobi equation with the form S ( x,u ) = 1 2 ( Ax 2 + 2 Bxu + Cu 2 ) . Solution Using S x = Ax + Bu and S u = Bx + Cu , the Hamilton- Jacobi equation becomes 0 = S x + H ( x,u,S u ) = Ax + Bu- q x 2 + u 2- ( Bx + Cu ) 2 . Solve this to find that A =- C and B = √ 1- A 2 . A general solution of the given form is then S ( x,u ) = 1 2 Ax 2 + p 1- A 2 xu- Au 2 ....
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This note was uploaded on 09/09/2009 for the course MATH 410 taught by Professor Staff during the Spring '08 term at Maryland.
- Spring '08