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Unformatted text preview: Physics 580 Handout 4 24 August 2010 Quantum Mechanics I webusers.physics.illinois.edu/ goldbart/ Homework 0 Prof. P. M. Goldbart 3135 (& 2115) ESB University of Illinois This particular homework assignment is optional , but I encourage you to work through at least some of the problems, especially numbers 2 , 8 , and 10 . 1) Calculus of variations straight lines : a) By using the calculus of variations with fixed end-points, show that the shortest path between two fixed points in a plane is a straight line. b) By using the calculus of variations with fixed end-points, for the case of several depen- dent variables, show that the shortest path between two points in three dimensional space is given by a straight line. c) Reconsider the derivation of the Euler equation for a single dependent variable, now allowing variations that do not necessarily vanish at the ends. By paying particular attention to the boundary terms, show that in the plane the shortest path between a straight line and a point is the perpendicular from the point to the line. [Note: For the purposes of this question you need only establish that such distances are stationary with respect to small variations, and not necessarily (even local, let alone global) minima.] 2) The brachistochrone problem : In this question we will use the calculus of variations to solve a famous old problem, the brachistochrone (or least time) problem. Let A and B be two points in a vertical plane, with A higher than B . Suppose that A and B are connected by a smooth wire that describes some curve in the vertical plane, and that the downward gravitational acceleration is g . A bead starts from rest at the point A and is allowed to slide to the point B . T , the time taken to slide from A to B , depends on the shape of the curve. By following the steps outlined below, find the curve that makes T stationary. a) Let y be the depth below A and let x be the horizontal displacement from A . Then A is the point ( x, y ) = (0 , 0), and B is the point ( x,y ) = ( x, y ). Construct a functional T , which depends on g and on the curve of the wire y ( x ), that gives the time taken to reach B . Write T in the form T [ y ] = 1 2 g x dxf ( y, y ) , where y denotes dy/dx , and write down the appropriate function f . b) Derive the Euler-Lagrange equation for the curve y ( x ) that makes T stationary. Of what order is this differential equation? 1 c) The solution of this problem is simplified by recognising that the function f does not depend on x . To see the simplification, consider a general functional of the form U [ y ] = x dx h ( y, y ) . Write down the Euler-Lagrange equation for this problem, in terms of the function h ( y, y ). Show that, because h/x = 0, we can immediately integrate the Euler- Lagrange equation to obtain the so-called first integral h y h y = constant ....
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