Solutions to a Modified Newtonian Dynamics Force Law

We consider a specific relation for a modified Newtonian force law\footnote{M. Milgrom, Astrophysical J., Vol. 270 (1983), 365.} in one space dimension: \[ \frac{m|a|a}{(a_0+|a|)}=F(x),\] where $a=d^2x/dt^2$ and $a_0$ is a very small ``cosmic'' related acceleration.\footnote{J. H. Gundlach, et al., Phys. Rev. Letters. Vol. 98 (2007), 150801.} Exact solutions are calculated for zero, constant, and linear damping forces. However, the linear harmonic oscillator force situation could not be solved exactly and, as a consequence, an approximation to the periodic solutions was determined. To check the consistency of the calculations, we took the $a_0\to 0$ limits and found that the prior known results were obtained for all four systems.