  If a dynamical system is long-lived and non-resonant (that is, if there is a
set of tracers that have evolved independently through many orbital times), and
if the system is observed at any non-special time, it is possible to infer the
dynamical properties of the system (such as the gravitational force or
acceleration law) from a snapshot of the positions and velocities of the tracer
population at a single moment in time. In this paper we describe a general
inference technique that solves this problem while allowing (1) the unknown
distribution function of the tracer population to be simultaneously inferred
and marginalized over, and (2) prior information about the gravitational field
and distribution function to be taken into account. As an example, we consider
the simplest problem of this kind: We infer the force law in the Solar System
using only an instantaneous kinematic snapshot (valid at 2009 April 1.0) for
the eight major planets. We consider purely radial acceleration laws of the
form a_r = -A [r/r_0]^{-\alpha}, where r is the distance from the Sun. Using a
probabilistic inference technique, we infer 1.989 < \alpha < 2.052 (95 percent
interval), largely independent of any assumptions about the distribution of
energies and eccentricities in the system beyond the assumption that the system
is phase-mixed. Generalizations of the methods used here will permit, among
other things, inference of Milky Way dynamics from Gaia-like observations.
