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.
The Vlasov system is a transport equation describing the free flow of collisionless particles. Let 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 denote proper time as experienced by the particle, and the world-line of the particle (its spacetime trajectory) parametrized by , then we have the hyperbolic system of equations.