Geodesics As Hamiltonian Flows - Overview

Overview

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 with p being the momentum. It is the conservation of momentum that leads to the straight motion of a particle. On a curved surface, exactly the same ideas are at play, except that, in order to measure distances correctly, one must use the metric. To measure momenta correctly, one must use the inverse of the metric. The motion of a free particle on a curved surface still has exactly the same form as above, i.e. consisting entirely of a kinetic term. The resulting motion is still, in a sense, a "straight line", which is why it is sometimes said that geodesics are "straight lines in curved space". This idea is developed in greater detail below.