Equivalence Between Lagrangian And Hamiltonian Systems At The Post-Newtonian

It was claimed recently that a low order post-Newtonian(PN) Lagrangian formulation, Whose Euler-Lagrange equations are up to an infinite PN order, can be identical to a PN Hamiltonian formulation at the infinite order from a theoretical point of view. In general, this result is difficult to check because the detailed expressions of the Euler-Lagrange equations and the equivalent Hamiltonian at the infinite order are clearly unknown. However, there is no difficulty in some cases.In fact, this claim is shown analytically by means of a special first-order post-Newtonian(1PN) Lagrangian formulation of relativistic circular restricted three-body problem, where both the Euler-Lagrange equations and the equivalent Hamiltonian are not only expanded to all PN order,but have converged functions.It is also shown numerically that both the Euler-Lagrange equations of the low order Lagrangian and the Hamiltonian are equivalent only at high enough finite order.