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Unformatted text preview: Physics 847: Problem Set 6 Due Thursday, May 20, 2010 at 11:59 P. M. Each problem is worth 10 points unless specified otherwise. 1. Pathria, Problem 11.7 2. (25 pts.) Consider a one-dimensional chain of particles connected by springs. Let the mass of each particle be m , and let each spring constant be K . Suppose that the equilibrium length of each spring is a , and let the position of the n th mass be denoted na + u n . We assume that there are a total of N masses with periodic boundary conditions, so that u N +1 = u 1 . The masses are constrained to vibrate only along the chain. Thus, the Hamiltonian of the system may be written H = N summationdisplay n =1 bracketleftBigg p 2 n 2 m + K 2 ( u n +1- u n ) 2 bracketrightBigg . (1) (a). Write down the set of N coupled second-order equations of mo- tion for the u n s, either by solving Hamiltons equations or by using Newtons laws directly....
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