Note on the Equivalence of Post-Newtonian Lagrangian and Hamiltonian Formulations

Recently, it has been generally claimed 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 orders, 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 orders.