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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online