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Unformatted text preview: Theoretical Dynamics October 14, 2010 Homework 6 Instructor: Dr. Thomas Cohen Submitted by: Vivek Saxena Problem 1 Part (a) The equation of motion is d d (( m + S ) u ) = S (1) which can be rewritten as d d ( mu ) = S d d ( S u ) = S d S d u + S du d = S S u u  S du d (2) where we have used the fact that d S /d = u g S = u S . Part (b) Multiplying both sides of the above equation by u we get m du d u = u S S u u u  S du d u = u S S u  S du d u (as u u = 1) which implies ( m + S ) du d u = 0 (3) As S is a spacetime dependent scalar field, it is not identically equal to m and hence this implies du d u = 0 (4) 6 1 Part (c) The action is S = Z d ( m + S ) (5) = Z dt p 1 v 2 ( m + S ) (6) = Z dtL (7) So the Lagrangian is L = p 1 v 2 ( m + S ) (8) For S << m and nonrelativistic conditions, L = p 1 v 2 ( m + S )  m m 2 v 2 + S + O ( v 4 ) (9) 1 2 m ~x 2 S m where ~x = v (10) So the Lagrangian equals 1 2 m ~x 2 S up to an irrelevant constant m , which does not affect the equations of motion. The equations of motion in this regime are t L x i = L x i which, from equation (10), can be written as the vector equation, m ~x = S (11) Problem 2 S = Z d ( m + A u ) (12) = Z dt p 1 v 2 ( m + A u ) as dt = d/ (13) = Z dtL (14) where L = p 1 v 2 ( m + A u ) (15) Now, L = p 1 v 2 ( m + A u ) = m p 1 v 2 + 1 ( A g u ) = m p 1 v 2 + 1 ( A t  A v ) = m p 1 v 2 + A t A j v j (16) 6 2 The EulerLagrange equations are d dt L x i = L x i (17) = d dt mv i 1 v 2 A i = A t x i A i x j v j = d dt mv i 1 v 2 = dA i dt + A t x i A i x j v j = d dt ( mu i ) = A i t + A i x j v j + A t x i A j x i v j = d d ( mu i ) = A i t + A i x j v j + A t x i  A j x i v j = d d ( mu i ) = A i t + A i x j v j + A t x i  A j x i v j = d d ( mu i ) = A i x u  A x i u (18) = d d ( mu i ) = ( A i i A ) u (19) So the equation of motion is d d ( mu ) = A x  A x u (20) Problem 3 Part (a) For A to transform as a 4vector, we must have A  A = L A (21) where L is the Lorentz transformation. Now, under a gauge transformation, A  A = A + G (22) So, under a gauge transformation, the field tensor F is invariant, and hence the fields (which are derivable from the field tensor) are also invariant: F = A  A (23) = ( A + G )...
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This note was uploaded on 12/29/2011 for the course PHYSICS 601 taught by Professor Hassam during the Fall '11 term at Maryland.
 Fall '11
 Hassam
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