Abstract
In relativistic celestial mechanics, post-Newtonian (PN) Lagrangian and PN Hamiltonian formulations are not equivalent to the same PN order as our previous work in PRD (2015). Usually, an approximate Lagrangian is used to discuss the difference between a PN Hamiltonian and a PN Lagrangian. In this paper, we investigate the dynamics of compact binary systems for Hamiltonians and Lagrangians, including Newtonian, post-Newtonian (1PN and 2PN), and spin–orbit coupling and spin–spin coupling parts. Additionally, coherent equations of motion for 2PN Lagrangian are adopted here to make the comparison with Hamiltonian approaches and approximate Lagrangian approaches at the same condition and same PN order. The completely opposite nature of the dynamics shows that using an approximate PN Lagrangian is not convincing. Hence, using the coherent PN Lagrangian is necessary for obtaining an exact result in the research of dynamics of compact binary at certain PN order. Meanwhile, numerical investigations from the spinning compact binaries show that the 2PN term plays an important role in causing chaos in the PN Hamiltonian system.
Highlights
The first detection of gravitational waves by LIGO and Virgo confirmed Einstein’s (1915s) theory of general relativity [1,2,3,4,5]
We have studied the effect of spin–spin couplings on the chaos of the double black hole system when that is of the 2PN order [29]
That means for the PN Lagrangian formulation, where harmonic coordinates are used, the approximation is mainly caused by the substitution of the lower-order acceleration terms for higher-order ones; this is not the case in the treatment of the PN Hamiltonian system
Summary
The first detection of gravitational waves by LIGO and Virgo confirmed Einstein’s (1915s) theory of general relativity [1,2,3,4,5]. Lagrangian approaches with a single spinning body can present chaos in [42,46], whereas the Arnowitt–Deser–Misner (ADM) 2PN Hamiltonian two-black hole was integrable and non-chaotic [42,49]. Under the same coordinate gauge, there are differences between the post-Newtonian Hamiltonian and Lagrangian forms of the same order, as we investigated in previous work. For certain initial conditions of compact binary systems with comparable mass, the conservative PN Hamiltonian and Lagrangian forms are both dynamically chaotic [54]. We investigated the PN Hamiltonian dynamics of spinning compact binaries, which contained the 1PN, 2PN, and 3PN order spin–orbit couplings They are all linear functions of spins and momenta due to the absence of the 1PN term.
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