### Newton-Cartan part 3: gravitating particles

As a simple example of a physical theory on a Galilean manifold, let us consider the physics of a collection of massive particles that do not interact except for their gravitational interaction. In other words, let us consider a collisionless kinetic theory coupled to Newtonian gravity.

Vlasov system
The Vlasov system is a transport equation describing the free flow of collisionless particles. Let $(M,\nabla)$ be a manifold with an affine connection that represents the spacetime. We postulate Newton’s first law:

Physical assumption 1
The motion of a free particle is geodesic.

Therefore the motion of a free particle is described by the following system of equations: let $\tau$ denote proper time as experienced by the particle, and $\gamma(\tau)$ the world-line of the particle (its spacetime trajectory) parametrized by $\tau$, then we have the hyperbolic system of equations.

Equation 2
$(\frac{d}{d\tau}\gamma)(\tau) = V\circ\gamma(\tau) \in T\gamma \subset TM$
and
$\frac{d}{d\tau}(V\circ\gamma) (= \frac{d^2}{d\tau^2}\gamma) = \nabla_VV = 0$