2005AugQualMATH - Ben Sauerwine Practice for Qualifying...

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Ben Sauerwine Practice for Qualifying Exams Mathematical Methods Problem Source: CMU August 2005 Qualifying Exam Suppose we are interested in studying the motion of an an-harmonic oscillator whose Lagrangian is given by () 3 2 2 2 1 2 1 x t g kx x m L + = & we will assume that g(t) is small and acts over time as t δ which is much shorter than the time scale k m = 0 1 ω . That is, we will treat the tri-linear term as a perturbation. We wish to find the motion of the oscillator to linear order in g at a time well after the perturbation is turned off. To do this we will use the Green’s function method. (a) Write down the equations of motion. 2 3 x t g kx x m x L x L dt d + = = & & & (b) The Green’s function obeys ()( ' ' ' 2 0 t t t t G t t G = + & & ) (*) Show that the motion of x is given by () () ( ) ( + = ' ' ' ' 3 2 0 t x t g t t G dt m t x t x ) where is a solution to the simple harmonic oscillator motion, (i.e., ignoring the anharmonic term.) t x 0 Specifically, + ' ' ' ' ' 2 0 2 2 dt
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This note was uploaded on 04/07/2012 for the course PHYSICS 767 taught by Professor Dr.jaouni during the Spring '12 term at Abu Dhabi University.

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2005AugQualMATH - Ben Sauerwine Practice for Qualifying...

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