Geodesics As Hamiltonian Flows - Hamiltonian Approach To The Geodesic Equations

Hamiltonian Approach To The Geodesic Equations

Geodesics can be understood to be the Hamiltonian flows of a special Hamiltonian vector field defined on the cotangent space of the manifold. The Hamiltonian is constructed from the metric on the manifold, and is thus a quadratic form consisting entirely of the kinetic term.

The geodesic equations are second-order differential equations; they can be re-expressed as first-order ordinary differential equations taking the form of the Hamiltonian–Jacobi equations by introducing additional independent variables, as shown below. Start by finding a chart that trivializes the cotangent bundle T∗M (i.e. a local trivialization):

where U is an open subset of the manifold M, and the tangent space is of rank n. Label the coordinates of the chart as (x1, x2, …, xn, p₁, p₂, …, p_n). Then introduce the Hamiltonian as

Here, gab(x) is the inverse of the metric tensor: gab(x)g_bc(x) = . The behavior of the metric tensor under coordinate transformations implies that H is invariant under a change of variable. The geodesic equations can then be written as

and

The second order geodesic equations are easily obtained by substitution of one into the other. The flow determined by these equations is called the cogeodesic flow. The first of the two equations gives the flow on the tangent bundle TM, the geodesic flow. Thus, the geodesic lines are the projections of integral curves of the geodesic flow onto the manifold M. This is a Hamiltonian flow, and that the Hamiltonian is constant along the geodesics:

Thus, the geodesic flow splits the cotangent bundle into level sets of constant energy

for each energy E ≥ 0, so that

.