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AbrahamLorentz-b - Supplement Phys411Spring 2010 Prof Ted...

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Supplement —Phys411—Spring 2010 Prof. Ted Jacobson Room 4115, (301)405-6020 www.physics.umd.edu/grt/taj/411b/ [email protected] Runaway solutions and pre-acceleration With the radiation reaction force F rad = ( μ 0 q 2 / 6 πc a , the equation of one dimensional motion in the presence of an external force F ( t ) takes the form a - τ ˙ a = f ( t ) , (1) where τ = μ 0 q 2 / 6 πcm , and f ( t ) = F ( t ) /m , where m is the mass of the particle. This is equation is third order in time derivatives of position, so a solution is determined by initial position, velocity and acceleration. In terms of a ( t ) however it is first order, and can be solved with one undetermined constant. Given that solution, the position can be found given an initial position and velocity. Note that the equation does not have time reversal symmetry. To solve (1), let a ( t ) = e t/τ a 1 ( t ), so a - τ ˙ a = - τe t/τ ˙ a 1 . In terms of a 1 , (1) takes the form ˙ a 1 = - τ - 1 e - t/τ f ( t ) , (2) which has the general solution a 1 ( t ) = a 1 ( ) + τ - 1 Z t e - t 0 f ( t 0 ) dt 0 . (3)
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