05_580_hw_1 - Physics 580 Quantum Mechanics I Prof. P. M....

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Handout 5 3 September 2010 Quantum Mechanics I webusers.physics.illinois.edu/ goldbart/ Homework 1 Prof. P. M. Goldbart University of Illinois This homework is due to be handed in to the PHYS 580 Homework Box by 2:30 p.m. on Thursday September 9. 1) Lagrangian for a compound plane pendulum : Consider a pendulum comprising an upper segment, of length L 1 and bob mass M 1 , and a lower segment that is freely hinged to the bob of the upper segment, of length L 2 and bob mass M 2 . The segments move in a common vertical plane in the presence of gravity (acceleration g ). Let the angular displacements of the upper and lower segments from the vertical direction be, respectively, θ 1 and θ 2 . a) Construct a Lagrangian for the compound plane pendulum in terms of θ 1 and θ 2 . b) Hence, obtain the equations of motion for the compound plane pendulum. c) By linearizing about the position of mechanical equilibrium, determine the frequen- cies of the normal modes of oscillation of the compound pendulum in terms of its parameters, 2) Stephenson-Kapitza pendulum : This type of pendulum is a simple pendulum of length L and natural frequency ω , except that the point of support is forced to vibrate. In the simplest case, which we consider here, the point of support undergoes vertical harmonic oscillations of frequency Ω ω and amplitude qL . a) Construct the Lagrangian for the Stephenson-Kapitza pendulum, choosing as the co- ordinate the angular displacement θ away from the downward vertical. b) Hence, obtain the equation of motion for the Stephenson-Kapitza pendulum. What is striking about the Stephenson-Kapitza pendulum is that if the support is vibrated sufficiently rapidly (i.e., for q > 2), the pendulum has a stable configuration when it is inverted (i.e., when θ = π ), about which it can undergo stable, small oscillations at a frequency given by ω 1 + 1 2 q 2 2 . For further details, see E.I. Butikov, On the dy- namic stabilization of an inverted pendulum , Am. J. Phys. 69 (2001), 755-768; and/or http://www.fam.web.mek.dtu.dk/FVP/Jon/Theory1.html . The method of multiple-scale analysis provides a mathematically precise approach to this problem; for this and many other powerful techniques, see, e.g., C.M. Bender and S.A. Orszag,
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05_580_hw_1 - Physics 580 Quantum Mechanics I Prof. P. M....

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