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Unformatted text preview: Physics 318: Problem Set 3 Due Wednesday, Feb 13, 2008 1. Consider a particle of mass m and charge q acted on by electric and magnetic fields E ( x ,t ) and B ( x ,t ). These fields can be described in terms of the scalar and vector poten- tials Φ( x ,t ) and A ( x ,t ), for which E = −∇ Φ − ˙ A and B = ∇ × A . Take the generalized coordinates to be the components ( x,y,z ) = ( x 1 ,x 2 ,x 3 ) of the particle’s position, and take the Lagrangian to be L ( x i , ˙ x i ,t ) = 1 2 m ˙ x 2 − q Φ( x ,t ) + q ˙ x · A ( x ,t ) . a. Show that the Lagrange equations of motion for this Lagrangian give back the Lorentz force law m ¨ x = q E + q ˙ x × B . b. Show that the Hamiltonian H = ∑ i p i ˙ x i − L , where p i = ∂ L /∂ ˙ x i , is H = 1 2 m ˙ x 2 + q Φ( x ,t ) . Note that this differs from the sum T + U of kinetic and potential energies. 2. Consider a system of N particles with positions r n ( t ) for 1 ≤ n ≤ N . Suppose that the system is subject to the holonomic constraints...
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