p09fl15 - Lecture 15: Oscillators II (7 Oct 09) review 5...

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Lecture 15: Oscillators II (7 Oct 09) review 5 Oct HW A. Review 1. Expansion of both kinetic energy and potential energy in departures from equilibrium positions x (0) j , so derivatives are evaluated at q (0) j T = X j m j 2 ˙ x 2 j = X j m j 2 [ X i ∂x j ∂q i ˙ q i ] 2 1 2 X ij T ij ˙ q i ˙ q j ΔΦ = 1 2 X ij Φ ij q i q j where T and Φ ij are real symmetric matrices, T ij = T ji and Φ ij = Φ ji . Because the kinetic energy is intrinsically a positive quantity, the matrix T ij must be positive definite. Also Φ ij must be positive definite for stable oscillations, but that is not automatic for arbitrary expansion point. 2. The Lagrange equations then become: X j T ij ¨ q j = - X j Φ ij q j 3. Look for oscillator solutions in form δq j = A j exp( - ıωt + φ j ) = q 0 j exp( - ıωt ) 4. Then the Lagrange equations are ω 2 X j T ij q 0 j = X j Φ ij q 0 j 5. This has the form of a generalized eigenvalue problem λ T · q = M · q 1
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Theorems that λ and the eigenvectors are real carry over to this form,
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p09fl15 - Lecture 15: Oscillators II (7 Oct 09) review 5...

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