# p09fl14 - Lecture 14 Small oscillations(5 Oct 09 A Small...

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Lecture 14: Small oscillations (5 Oct 09) A. Small amplitude expansion 1. FW Sec. 21 starts with 1D picture (expand about a position of zero force) and then to quadratic forms for expansion of both kinetic energy and potential energy: 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 where T ij = T ji (if not seen immediately, juggle the dummy indices) [ T must be a positive deﬁnite matrix, from its role in the (positive) kinetic energy]. Then with the q i being increments relative to a position with zero gradients: ΔΦ = 1 2 X ij Φ ij q i q j The matrix Φ ij must be positive deﬁnite [ x T · Φ · x > 0] for stable oscillations but that is not automatic for expansion about an arbitrary point. 2. Cautionary remark for molecular vibrations (e.g., triatomic molecule). Expand the potential Φ( r ij ) relative to its equilibrium length r 0 using r i j = r 0 ˆ n + δ r ij : r ij r 0 + ˆ n · δ r ij Φ( r ij ) Φ( r 0 ) + 1 2 n · δ

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## This note was uploaded on 12/14/2009 for the course PHYS 711 taught by Professor Bruch during the Fall '09 term at University of Wisconsin.

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p09fl14 - Lecture 14 Small oscillations(5 Oct 09 A Small...

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