In the article from Wikipedia called: Geodesics as Hamiltonian Flows at: http://en.wikipedia.org/wiki/Geodesics_as_Hamiltonian_flows" [Broken] It states the following: It is frequently said that geodesics are "straight lines in curved space". By using the Hamilton-Jacobi approach to the geodesic equation, this statement can be given a very intuitive meaning: geodesics describe the motions of particles that are not experiencing any forces. In flat space, it is well known that a particle moving in a straight line will continue to move in a straight line if it experiences no external forces; this is Newton's first law. The Hamiltonan describing such motion is well known to be H = mv2 / 2 = p2 / 2m with p being the momentum. It is the conservation of momentum that leads to the straight motion of a particle. Under the wiki article regarding momentum it states: Conservation of momentum is a mathematical consequence of the homogeneity (shift symmetry) of space (position in space is the canonical conjugate quantity to momentum). So, momentum conservation can be philosophically stated as "nothing depends on location per se". My understanding of relativity is fairly basic but I feel I intuitively understand most of this. My question is if we insert the second bold text into the first it basically says that straight line motion is a result of shift symmetry. Can someone explain this further? Or am I fishing for a connection between two related but ancillary points?