set6 - Department of Physics The Ohio State University...

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Department of Physics Prof. R. Bundschuh The Ohio State University Sixth Problem Set for Physics 664 (Theoretical Mechanics) Spring quarter 2004 Important dates: May 11 9:30am-10:18am midterm exam, May 31 no class, Jun 9 9:30am-11:18am Fnal exam Due date: Thursday, May 13 15. Particle on a helix 9 points A particle of mass m moves under the in±uence of gravity along the helix z = , r = constant, where k is a constant and the z -axis points vertically upwards. a) Write down the Lagrangian of the system expressing it in the variable z . b) Calculate the generalized momentum p z . c) Calculate the Hamiltonian in terms of the variables z and p z . d) Derive the Hamiltonian equations of motion of the system. e) Solve the Hamiltonian equations of motion. 16. Hamiltonians and energy 22 points In this problem we want to develop a feeling for the relationship between the Hamiltonian and the total energy of a system and their conservation. To this end, write down the kinetic energy, the potential energy, the total energy, the Lagrangian, the generalized momenta, and
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