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

• Published in: Communications in theoretical physics Vol. 65; no. 3; pp. 321 - 328
• Main Author: 陈荣超 伍歆
• Format: Journal Article
• Language: English
• Published: 01.03.2016
• Subjects: Circularity; Equivalence; Euler-Lagrange equation; Hamilton; Hamiltonian functions; Mathematical analysis; Mathematical models; Theoretical physics; 拉格朗日公式; 欧拉-拉格朗日方程; 注记; 牛顿方法; 等价性; 等效哈密顿量; 限制性三体问题
• ISSN: 0253-6102; 1572-9494
• DOI: 10.1088/0253-6102/65/3/321

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.