Solutions to the Equation of Motion

Abstract

AbstractThis chapter begins by solving the equation of motion for the extended charge in rectilinear motion to obtain the well-known pre-acceleration solution. The root cause of the noncausal pre-acceleration is traced to the assumption in the classical derivation of the self electromagnetic force that the position, velocity, and acceleration of each element of charge at the retarded times can be expanded in a Taylor series about the present time. With a finite external force that is zero for all time less than zero and yet an analytic function of time about the real t axis for all time greater than zero, these Taylor series expansions are valid for all time except during the initial short time interval light takes to traverse the charge distribution. If the derivation of the self force is done properly near the initial time, an effective correction force that is nonzero only in this short “transition interval” must be included in the equation of motion. This transition force in the equation of motion removes the noncausal pre-acceleration from the solution to the equation of motion without destroying the covariance of the equation of motion. When the corrected equation of motion is applied to the problem of determining the motion of a charge that is accelerated by a uniform electric field between the parallel plates of a charged capacitor, the addition of transition forces at the two nonanalytic points of time in the external force (one when the charge enters the first plate and one when it exits the second plate) eliminates both pre-acceleration and pre-deceleration.If one is not concerned with the correct behavior of the solution to the equation of motion during the time immediately after the external force is first applied, one can obtain a convenient power series solution to the equation of motion, in particular, the Landau-Lifshitz approximate solution to the Lorentz-Abraham-Dirac equation of motion. For the special case of a charge moving in a uniform magnetic field, the solution is given for the Landau-Lifshitz approximation to the Lorentz-Abraham-Dirac equation of motion. A three-vector formulation of the Landau-Lifshitz approximation is also derived in closed-form for an electron encountering a counterpropagating plane-wave laser pulse. General conditions are obtained for the Landau-Lifshitz approximation to be an accurate solution to the Lorentz-Abraham-Dirac equation of motion, and for the Landau-Lifshitz solution to reduce to the Lorentz-force solution.Simple semi-classical derivations are used to obtain the parameters and conditions for deciding the importance of both quantum Compton scattering of the electron by the incident photons and the electron quantum recoil effects produced by the emitted photons. It is proven that the Landau-Lifshitz approximation becomes an inaccurate solution to the Lorentz-Abraham-Dirac equation of motion only where the product of the laser intensity and the relativistic factor is large enough that quantum recoil effects on the electron can dominate the classical solution.The finite difference equation of motion is derived for the extended charge. We find that there is little justification to accept the finite difference equation as a valid equation of motion because it neglects all nonlinear terms (in the proper frame of the charge) involving products of the time derivatives of the velocity, and retains a homogeneous runaway solution that leads to pre-acceleration.Lastly we concentrate on the effect of renormalizing the mass of the electron to its finite value as the radius of the extended charge approaches zero to obtain the Lorentz-Abraham-Dirac equation of motion. Although renormalization is an extremely simple, seemingly innocuous way to ensure the correct measured rest mass in the classical equation of motion of a point charge, it is not an entirely seamless alteration.KeywordsCharge in uniform magnetic fieldElectron in laser beamFinite difference equation of motionLandau-Lifshitz solutionLorentz-Abraham-Dirac equation of motionParallel-plate capacitorPre-accelerationPre-decelerationQuantum effectsRenormalizationTransition forces