ps4 - center so that it slides down the hemisphere without...

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PHY 504 Problem Set #4 due 24 September 2010 1. Goldstein Exercise 2.12. Show that the Lagrangian at the end of the problem differs from a more familiar Lagrangian by a term which is a total time derivative. Assuming the result of Derivation 1.8, can you explain why the equations of motion have the form that they do? 2. A particle of mass m slides in a frictionless trough of parabolic shape y = kx 2 , under the influence of a gravitational force F = − mg ˆ y . (a) Write down a Lagrangian for the particle’s motion, using the method of Lagrange multipliers. (b) Lag and constraint. Eliminate y and ge a a ion just involving x ( t ). Obtain the range equations of motion the Lagran multiplier λ to obtain n equ t (c) Solve for x ( t ) in the limit of small x and . x & 3. A particle slides on top of a smooth solid, immovable hemisphere of radius a, in a gravitational field . It is initially given an infinitesimal displacement off
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Unformatted text preview: center, so that it slides down the hemisphere without friction. Describe the subsequent motion of the particle. [See p. 47 of GPS.] y ˆ g − 4. Write down a Lagrangian for each of the following situations, in terms of an appropriate set of generalized coordinates. Use Lagrange multipliers as needed. What are the symmetries? Give expressions for the corresponding conserved quantities. (a) A particle of mass m moving on the frictionless surface of sphere. (b) Two particles constrained to move on the surface of a sphere, interacting gravitationally with each other. (c) A charged particle moving in a uniform electric field. (d) A charged particle moving in a uniform magnetic field. In this case, use Noether’s Theorem to derive the conserved momenta, angular momentum (in the direction parallel to the magnetic field), and energy....
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