How to Find a Closed Trajectory for a Wormhole Metric?

[vilasini]

Homework Statement

I have a metric for wormhole. Now if I want to find the closed trajectory for this metric how will I proceed from here?
Help me with detailed maths if possible.

Homework Equations

The Attempt at a Solution

 

[member 553083]

The closed trajectory for a metric can be found using the Hamilton-Jacobi equation. This equation states that the Hamiltonian of the system is equal to the partial derivative with respect to the action of the Hamilton-Jacobi function. The Hamilton-Jacobi equation can be written as follows(x,p) = ∂S/∂t where x represents the coordinates of the system and p represents the momentum of the system. By solving this equation we can find the equation of motion of the system which will give us the closed trajectory of the system.