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This 2 page Class Notes was uploaded by Dr. Simeon Wiza on Wednesday September 9, 2015. The Class Notes belongs to PHYS 505 at University of Washington taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/192447/phys-505-university-of-washington in Physics 2 at University of Washington.
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Date Created: 09/09/15
Lecture 8 Small Oscillations Chapter 4 in F amp W We want to discuss the small oscillations that a mechanical system exhibits when perturbed around an equilibrium point as has already appeared in the HW exercises and the appendix to the previous lecture As usual we are imagining that the systems can be described by potentials U 611 61f and Lagrangians Lq1 q f 11 f We consider systems that exhibit equilibrium points in phase space where both 4 0 and pk 0 all k If we expand the behavior in terms of small perturbations about this point we expect to have a linear description of the system in which case we can always solve for the behavior of the system in terms of the normal modes the eigenfunctions with generic time dependence exp M II For expansions about a stable equilibrium a true minimum of the potential we find real values for the frequencies w For future reference we note that the corresponding flow in phase space recall the appendix to the last lecture is elliptical including circular For expansions about an unstable equilibrium a maximum or saddle point of the potential the frequency picks up an imaginary part and the system tends to run away from the initial point thereby lowering its energy This corresponds to hyperbolic ow in phase space We consider systems described by a positive kinetic energy defined in terms of a possibly coordinate dependent metric which includes any mass factors left over after the definition of q 1 TZEkalq17anfqkql 81 kl We are implicitly assuming that all conserved canonical momenta have been identified and replaced by their constant values in the effective potential which we will continue to call U ql q f The remaining coordinates are taken to be f in number Since T is positive the matrix mkl sometimes represented as gkl is both symmetric and positive definite Hence it also has an inverse m lld As usual we have Physics 505 Lecture 8 1 Autumn 2005
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