hw7 - March 12, 2010 Cornell University, Department of...

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March 12, 2010 Cornell University, Department of Physics P3318, Analytical Mechanics, HW#7, due: 3/19/10, 2:30PM (at section) Question 1 : Another 2 × 2 system Consider a particle of mass m moving in two dimensions such that V ( x 1 , x 2 ) = V 0 b cosh( ax 1 + bx 2 ) + cx 2 1 B . (1) 1. Expand the potential assuming small oscillations. Write the Lagrangian in a matrix notation as L = ij v i v j 2 k ij x i x j 2 i, j = 1 , 2 . (2) and write k ij explicitly. 2. Diagonalize k ij and Fnd the normal modes (that is, the mixing angle) and their fre- quencies. There is no need to simplify the expressions. 3. Write the solution, that is, x 1 ( t ) and x 2 ( t ), as a function of ω 1 , 2 , the mixing angle θ and the initial conditions which we take to be x 1 ( t = 0) = A 1 , x 2 ( t = 0) = A 2 , ˙ x 1 ( t = 0) = ˙ x 2 ( t = 0) = 0 . (3) 4. When is the small oscillation approximation valid? Write your answer as a condition of A 1 , A 2 and the Lagrangian parameters, or, equivalently, the normal frequencies. 5. Consider the case
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hw7 - March 12, 2010 Cornell University, Department of...

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