SOLUTION OF THE LIOUVILLE'S EQUATION FOR KEPLERIAN MOTION: APPLICATION TO UNCERTAINTY CALCULATIONS

摘要

In the absence of process noise, the evolution of uncertainty from one time step to another is governed by a partial differential equation called the stochastic Liouville's equation. It differs from the Fokker-Planck Kolmogorov equation by the fact that there is no diffusion in the evolution process. Being a first order, linear, partial differential equation in n-dimensions, the Liouville's equation in several cases admits exact solutions. In general problems, the method of characteristics is employed to obtain solution density functions to this equation. It is shown in this paper that an application of the transformation of variables formula from probability theory yields an exact solution. It is also shown that this is identical to using the method of characteristics, appealing to the fact that the characteristic curves are automatically obtained by using the solution trajectories. For the special case of Keplerian motion, an analytic expression governing the probability density function evolution is derived. It is shown that by using the Kepler elements, the solution process is simplified significantly.