CHASSIS - Inverse Modelling of Relaxed Dynamical Systems

Abstract

The state of a non-relativistic gravitational dynamical system is known at any time $t$ if the dynamical rule, i.e. Newton's equations of motion, can be solved; this requires specification of the gravitational potential. The evolution of a bunch of phase space coordinates ${\bf w}$ is deterministic, though generally non-linear. We discuss the novel Bayesian non-parametric algorithm CHASSIS that gives phase space $pdf$ $f({\bf w})$ and potential $\Phi({\bf x})$ of a relaxed gravitational system. CHASSIS is undemanding in terms of input requirements in that it is viable given incomplete, single-component velocity information of system members. Here ${\bf x}$ is the 3-D spatial coordinate and ${\bf w}={\bf x+v}$ where ${\bf v}$ is the 3-D velocity vector. CHASSIS works with a 2-integral $f=f(E, L)$ where energy $E=\Phi + v^2/2, \: v^2 = \sum_{i=1}^{3}{v_i^2}$ and the angular momentum is $L = |{\bf r}\times{\bf v}|$, where ${\bf r}$ is the spherical spatial vector. Also, we assume spherical symmetry. CHASSIS obtains the $f(\cdot)$ from which the kinematic data is most likely to have been drawn, in the best choice for $\Phi(\cdot)$, using an MCMC optimiser (Metropolis-Hastings). The likelihood function ${\cal{L}}$ is defined in terms of the projections of $f(\cdot)$ into the space of observables and the maximum in ${\cal{L}}$ is sought by the optimiser.