The closed trajectory for a wormhole metric can be determined using the Hamilton-Jacobi equation, which relates the Hamiltonian of the system to the partial derivative of the Hamilton-Jacobi function. Specifically, the equation is expressed as H(x,p) = ∂S/∂t, where x denotes the coordinates and p signifies the momentum. Solving this equation yields the equations of motion necessary to identify the closed trajectory of the system.
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(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.