The Irish physicist, Joseph Larmor, published in 1895, the first analysis conerning radiation by an accelerated charge based upon the derivation by Hendrik Lorentz of the force experienced by a particle due its charge field being dragged through the ether. It was not appreciated at the time but Larmor's analysis presumed that the velocity of the charge was small relative to the speed of light in a vacuum. Larmor's analysis in 1895 was followed by a more comprehensive analysis by Alfred-Marie Liénard in 1898 and, independently, by Emil Wiechert in 1900. Their analyses were compatible with Einstein's Theory of Special Relativity published in 1905. Liénard and Wiechert based their analyses on the vector and scalar potentials of the electric field of a particle.

Explanation of Why Accelerating Charges
DO NOT Emit Electromagnetic Radiation

The Larmor Effect is derived for point particles and depends upon the square of the particle's charge. If a charge of Q is distributed over M points then the M points each radiate an amount proportional to (Q/M)² for a total of Q²/M. If M goes to infinity, as it would for a spatially distributed charge no matter how small the region of distribution, then the radiation goes to zero. Period.

This is why there is no radiation from an electron or proton revolving about their center of mass in an atom. Likewise there is no radiation from protons in a rotating nucleus. And similarly there is no radiation from quarks composing particles.

The radiation from beams of charged particles in synchrotrons and other particle accelerators can be accounted for by the beams passing through the magnet fields used to generate the circular paths for the beams.

For more on the history of the Larmor Effect and skeptism concerning is validity see The Larmor Effect.

In general, any effect on a point particle that depends upon the charge of the particle raised to a power greater than 1.0 is irrelevant for particles with spatially distributed charges. That charge may be electrical, magnetic or gravitational.

Conclusion

There has been a good deal of skepticism among physicists concerning radiation by accelerated charged particles in the absence of any external electrical or magnetic field. A spatially distributed charge does not radiate, even if it is accelerated or decelerated.