Geodesics As Hamiltonian Flows - Geodesics As An Application of The Principle of Least Action

Geodesics As An Application of The Principle of Least Action

Given a (pseudo-)Riemannian manifold M, a geodesic may be defined as the curve that results from the application of the principle of least action. A differential equation describing their shape may be derived, using variational principles, by minimizing (or finding the extremum) of the energy of a curve. Given a smooth curve

that maps an interval I of the real number line to the manifold M, one writes the energy

where is the tangent vector to the curve at point . Here, is the metric tensor on the manifold M.

Using the energy given above as the action, one may choose to solve either the Euler–Lagrange equations, or the Hamilton-Jacobi equations. Both methods give the geodesic equation as the solution; however, the Hamilton–Jacobi equations provide greater insight into the structure of the manifold, as shown below. In terms of the local coordinates on M, the (Euler–Lagrange) geodesic equation is

Here, the xa(t) are the coordinates of the curve γ(t) and are the Christoffel symbols. Repeated indices imply the use of the summation convention.