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Homework 6

Homework 6 - 1 S = d(m S d m S u = S where d g x d u = S S...

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Physics 601 Homework 6---Due Friday October 15 1. In class starting with the action S = d τ m S ( ) we used covariance to show that for a particle moving in a Lorentz scalar field d m + S ( ) u μ ( ) d τ = μ S where μ g μ ν x ν . a. Show that this can be rewritten in the form m + S ( ) d u μ ( ) d τ = μ S α S ( ) u α u μ b. Show that this equation of motion automatically satisfies the condition d u μ u μ ( ) d τ = 0 . This indicates that imposition of covariance yielded a self‐ consistent result that respects the condition u μ u μ = 1 . c. Show that in the non‐relativistic limit where all of the velocities are much less than the speed of light and S << m the Lagrangian for the system reduces to L = 1 2 m ˙ x 2 S plus an irrelevant constant and the equation of motion reduces to m ˙ ˙ x = S . 2. Start from the action S = d τ m + V μ u μ ( ) where A μ is a four vector field that depends on space‐time. Show that the equation of motion is d mu μ ( ) d τ = V μ x ν V ν x μ u ν . 3. In electro‐magnetism, one can write the scalar and vector potentials in a form that looks like a 4‐vector: A μ = Φ A x A y A z . Because one can make arbitrary gauge transformations A μ need not transform as a 4‐vector.

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Homework 6 - 1 S = d(m S d m S u = S where d g x d u = S S...

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