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Unformatted text preview: Physics 318: Problem Set 5 Due Wednesday, Feb 27, 2008 1. Suppose that two Lagrangians L and L ′ are related by L ′ ( q, ˙ q, t ) = L ( q, ˙ q, t ) + d dt f ( q, t ) (1) where f is some function and q is shorthand for ( q 1 , . . . , q f ). a. By computing the Euler-Lagrange equations for the Lagrangian L ′ , show that the two Lagrangians give the same equations of motion. b. Consider the case of a particle of charge q and mass m moving in electric and magnetic fields parame- terized by a scalar potential Φ and a vector potential A . The Lagrangian for this system is L ( x , ˙ x , t ) = 1 2 m ˙ x 2 − q Φ( x , t ) + q ˙ x · A ( x , t ) . An electromagnetic gauge transformation is a transformation of the form A → A + ∇ ψ , Φ → Φ − ˙ ψ , where ψ = ψ ( x , t ) is an arbitrary function. Show that under such a transformation the Lagrangian transforms as in Eq. (1), and compute the form of the function f . 2. Consider the Lagrangian L ( q 1 , q 2 , ˙ q 1 , ˙ q 2 ) = 1 2 m ( ˙ q 2 1 + ˙ q 2 2 ) − 1 2 mω 2 ( q 2 1 + q 2 2 ) , which describes two uncoupled harmonic oscillators of mass m and frequency ω . Show that this system is invariant under the symmetry operation q ( t ) → e iα q ( t ) , where q ( t ) ≡ q 1...
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