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Ph12a_s1_2009

# Ph12a_s1_2009 - Ph12a Solution Set 1 October 4 2009 1.15...

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Ph12a Solution Set 1 October 4, 2009 1.15 The equations of motion for two coupled oscillators can be put in the form d 2 ψ a dt 2 = - a 11 ψ a - a 12 ψ b (1) d 2 ψ b dt 2 = - a 21 ψ a - a 22 ψ b (2) Suppose we have two solutions to these equations, one for initial positions x a 1 and x b 1 , initial velocities v a 1 and v b 1 , and another with initial positions x a 2 and x b 2 , initial velocities v a 2 and v b 2 . That is, suppose ψ a 1 and ψ b 1 satisfy the equations of motion and ψ a 1 (0) = x a 0 ψ 0 a 1 (0) = v a 0 ψ b 1 (0) = x b 0 ψ 0 b 1 (0) = v b 0 and similarly for ψ a 2 and ψ b 2 . It is obvious that ψ a 3 = ψ a 1 + ψ a 2 and ψ b 3 = ψ b 1 + ψ b 2 will have the required initial conditions. What is needed to be seen is that these functions also satisfy (1) and (2), the equations of motion. This can also easily be seen as follows. d 2 ψ a 3 dt 2 = d 2 dt 2 ( ψ a 1 ( t ) + ψ a 2 ( t )) = d 2 ψ a 1 dt 2 + d 2 ψ a 2 dt 2 = ( - a 11 ψ a 1 - a 12 ψ b 1 ) + ( - a 11 ψ a 2 - a 12 ψ b 2 ) = - a 11 ( ψ a 1 + ψ a 2 ) - a 12 ( ψ b 1 + ψ b 2 ) = - a 11 ψ a 3 - a 12 ψ b 3 and similarly for eqn (2).

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