MATH503 hw3 - the general solution y ( x ) that minimizes E...

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Homeworks (Math 503) T1 : Div, grad, curl and all that , by Schey (4th edition) T2 : Calculus of variations T3 : Nonlinear dynamics and chaos , by Strogatz Note: Detail your work to receive full credit Homework#3: (due Wed Oct 20) 1. Find the extremals of (a) J ( y ) = Z π/ 2 0 ( y 0 2 - y 2 ) dx with y (0) = 3 / 2 and y ( π/ 2) = 1 / 2 (b) J ( y ) = Z b a y 0 2 x 3 dx with y ( a ) = y a and y ( b ) = y b (c) J ( y ) = Z π/ 2 0 ( y 2 + y 0 2 - 2 y sin x ) dx with y (0) = 0 and y ( π/ 2) = - cosh( π/ 2) / 2 2. A particle under the influence of a gravitational field moves on a path along which the kinetic energy E = m Z 1 0 q ( u 2 0 - 2 gy )(1 + y 0 2 ) dx is minimal, where m (mass), g (gravity) and u 0 (initial speed) are constant parameters. Find
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Unformatted text preview: the general solution y ( x ) that minimizes E given y (0) = 0. 3. Find the general solution of the Euler-Lagrange equation for J ( y ) = Z b a p x (1 + y 2 ) dx 4. Consider the functional J ( y ) = Z b a p ( x 2 + y 2 )(1 + y 2 ) dx (a) Use the polar coordinates with r being a function of to change J ( y ) to a functional in terms of r , r and . (b) Derive the corresponding Euler-Lagrange equation and simplify it as much as possible....
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This note was uploaded on 02/20/2011 for the course MATH 503 taught by Professor Schleiniger,g during the Fall '08 term at University of Delaware.

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