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PHYS 598 final f03 - This exam has four pages and six...

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Physics 498/MMA Handout FT Dec 20th 2002 Mathematical Methods in Physics I http://w3.physics.uiuc.edu/ m-stone5 Prof. M. Stone 2117 ESB University of Illinois This exam has four pages and six problems. Answer question one , and then any other three questions. Do not hand in solutions to more than this number of problems! Try to answer entire questions. Little, if any, credit will be given for fragmentary answers. Errors will not be propagated, so make sure of each step before you go on. 1) One-dimensional Green Function : Consider the differential equation - y 00 = f ( x ) with boundary conditions y 0 (0) = 0 and y (1) = 0. We are given f ( x ) and wish to solve for y ( x ). a) Construct the explicit Green function appropriate to this problem (5 points). b) Use your Green function to write down the solution of the boundary value problem as the sum of two explicit integrals over complementary components of the unit interval (5 points). c) Evaluate the x derivative of your solution, y ( x ), and confirm that y obeys both bound- ary conditions (5 points). d) Take one further derivative of your y ( x ) and confirm that it does indeed solve the original problem (5 points). 2) Bead and string : A bead of mass M is free to slide up and down the y axis. x y y(0) 0 L A bead connected to a string. It is attached to the x = 0 end of a string in such a way that the Lagrangian for the string-bead system is L = 1 2 M [ ˙ y (0)] 2 + Z L 0 1 2 ρ ˙
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