A 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...

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Published inCommunications in theoretical physics Vol. 65; no. 3; pp. 321 - 328
Main Author 陈荣超 伍歆
Format Journal Article
LanguageEnglish
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
Online AccessGet full text
ISSN0253-6102
1572-9494
DOI10.1088/0253-6102/65/3/321

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Summary: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.
Bibliography:post-Newtonian approximation; Lagrangian and Hamiltonian mechanics; circular restricted threebody problem; chaos
Rong-Chao Chen,Xin Wu（ Department of Physics and Institute of Astronomy, Nanchang University, Nanchang 330031, China）
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.
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ISSN:0253-6102
1572-9494
DOI:10.1088/0253-6102/65/3/321