is the position of the charge relative to the center of the circle and is the instantaneous velocity of the charge. The above Larmor formula becomes as follows: If we assume the motion of a charged particle is periodic, then the average work done on the particle by the Abraham–Lorentz force is the negative of the Larmor power integrated over one period from to : Notice that we can integrate the above expression by parts. If we assume that there is periodic motion, the boundary term in the integral by parts disappears: Integrating by parts a second time, we find Clearly, we can identify Signals from the future Below is an illustration of how a classical analysis can lead to surprising results. The classical theory can be seen to challenge standard pictures of causality, thus signaling either a breakdown or a need for extension of the theory. In this case the extension is to quantum mechanics and its relativistic counterpart quantum field theory. See the quote from Rohrlich in the introduction concerning "the importance of obeying the validity limits of a physical theory". For a particle in an external force , we have where This equation can be integrated once to obtain The integral extends from the present to infinitely far in the future. Thus future values of the force affect the acceleration of the particle in the present. The future values are weighted by the factor which falls off rapidly for times greater than in the future. Therefore, signals from an interval approximately into the future affect the acceleration in the present. For an electron, this time is approximately sec, which is the time it takes for a light wave to travel across the "size" of an electron. See also Max Abraham Hendrik Lorentz Cyclotron radiation Electromagnetic mass Radiation resistance Radiation damping Synchrotron radiation Wheeler–Feynman absorber theory References Further reading See sections 11.2.2 and 11.2.3 \ Jose A. Heras, The Radiation Force of an Electron Reexamined, 2003, http://www.joseheras.com/jheras_papers/JAH-PAPER_16.pdf. Donald H. Menzel, Fundamental Formulas of Physics, 1960, Dover Publications Inc., , vol. 1, page 345. External links MathPages - Does A Uniformly Accelerating Charge Radiate? Feynman: The Development of the Space-Time View of Quantum Electrodynamics Heras: The Radiation Reaction Force of an Electron Reexamined Electrodynamics Electromagnetic radiation