PHY4221 Homework Assignment 1 Solution (Written by: Leung Shing Chi) 1. (In this problem, we use Einstein’s summation convention, i.e., also we use the following notation:31iiiiiA AA A,ii jjAAx) Given the Lagrangian to be1( , )( , )2qLmv vqr tvA r tc , we apply the Euler-Lagrange Equation (ELE) to L: 0iidLLdtvxwhere 1,2,3i(3 equations in total). ,iiiijj iiiLqPmvAvcLqqvxxc ATherefore, we find ,iiiAdLdqmvv Adtvdtctji j(It should be noted that this result arises as the usage of ordinary derivative) and we take the ELE to be: ,,,,,01iiji jijiiiji jAdqqmvv Aqv AdtctcAdqmvqv Av Adtctc ,j ijj iThis, in fact, is the solution by observing the following relation among the scalar and vector potential and the Eand Bfields. ,,,1iiijkjkiim nmnkijkji jjj iiAEctvBv BvAv Av A ,Certainly, to achieve this result, we need some relations between and, namely: ijkmnkimjninjm Finally, it is now obvious to see the equation of motion (EoM) to be:
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