Second post-Newtonian Lagrangian dynamics of spinning compact binaries

Huang, Li; Wu, Xiaoxia; Ma, Da-Zhu

doi:10.1140/epjc/s10052-016-4339-7

Cited by 40 publications

(24 citation statements)

References 54 publications

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“…The equivalence between both approaches was shown in Damour et al (2001), Damour et al (2002), de Andrade et al (2001) and Levi & Steinhoff (2014). However, different claims exist in , , Wang & Huang (2015), Chen & Wu (2016) and Huang et al (2016), stating that such differences are owed to the truncation of higher-order PN terms.…”

Section: Introductionmentioning

confidence: 97%

On the conservation of the Jacobi integral in the post-Newtonian circular restricted three-body problem

Dubeibe

Lora-Clavijo

González

2017

Astrophys Space Sci

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In the present paper, using the first-order approximation of the n-body Lagrangian (derived on the basis of the post-Newtonian gravitational theory of Einstein, Infeld, and Hoffman), we explicitly write down the equations of motion for the planar circular restricted three-body problem. Additionally, with some simplified assumptions, we obtain two formulas for estimating the values of the mass-distance and velocity-speed of light ratios appropriate for a given post-Newtonian approximation. We show that the formulas derived in the present study, lead to a good numerical conservation of the Jacobi constant and allow for an approximate equivalence between the Lagrangian and Hamiltonian approaches at the same post-Newtonian order. Accordingly, the dynamics of the system is analyzed in terms of the Poincar sections method and Lyapunov exponents, finding that for specific values of the Jacobi constant the dynamics can be either chaotic or regular. Our results suggest that the chaoticity of the post-Newtonian system is slightly in-creased in comparison with its Newtonian counterpart.

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Section: Introductionmentioning

confidence: 97%

On the conservation of the Jacobi integral in the post-Newtonian circular restricted three-body problem

Dubeibe

Lora-Clavijo

González

2017

Astrophys Space Sci

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“…Chaos is a possibly terrible obstacle to a method of matched filtering, which requires a gravitational wave signal drawn out of the noise in excellent agreement with a theoretical template of the gravitational wave [4]. Thus, chaos in systems of spinning compact binaries has been studied by several authors [4][5][6][7][8][9][10][11][12][13][14]. The chaotic behavior in these references was mainly considered in conservative binary systems.…”

Section: Introductionmentioning

confidence: 99%

“…Hence the orbits of it are integrable and regular. The presence of chaos in [26,14,4,5,[7][8][9][10][11][12][13] arises due to the spins of the binary destroying the integrability of PN systems of compact binaries. The canonical, conjugate spin variables of Wu and Xie [33] play an important role in determining the integrability or nonintegrability of Hamiltonian systems of spinning compact binaries.…”

Section: Introductionmentioning

confidence: 99%

Extended phase-space symplectic-like integrators for coherent post-Newtonian Euler-Lagrange equations

Pan,

Wu,

Liang

2021

Preprint

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Coherent or exact equations of motion for a post-Newtonian Lagrangian formalism are the Euler-Lagrange equations without any terms truncated. They naturally conserve energy and angular momentum. Doubling the phase-space variables of positions and momenta in the coherent equations, we establish extended phase-space symplectic-like integrators with the midpoint permutations. The velocities should be solved iteratively from the algebraic equations of the momenta defined by the Lagrangian during the course of numerical integrations. It is shown numerically that a fourthorder extended phase-space symplectic-like method exhibits good long-term stable error behavior in energy and angular momentum, as a fourth-order implicit symplectic method with a symmetric composition of three second-order implicit midpoint rules or a fourth-order Gauss-Runge-Kutta implicit symplectic scheme does. For given time step and integration time, the former method is superior to the latter integrators in computational efficiency. The extended phase-space method is used to study the effects of the parameters and initial conditions on the orbital dynamics of the coherent Euler-Lagrange equations for a post-Newtonian circular restricted three-body problem. It is also applied to trace the effects of the initial spin angles, initial separation and initial orbital eccentricity on the dynamics of the coherent post-Newtonian Euler-Lagrange equations of spinning compact binaries.

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“…[6][7][8]. Due to higher-order terms truncated, the two formulations have somewhat differences and are not exactly equal [9][10][11]. For the Solar System as a weak gravitational field, the differences are too small to affect their equivalence, that is, the solutions of the two formulations should have no typical differences for the regular case.…”

Section: Introductionmentioning

confidence: 99%

The equations of motion for consistency of a post-Newtonian Lagrangian formulation

Li¹,

Wang²,

Deng³

et al. 2019

Preprint

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Equations of motion for a general relativistic post-Newtonian Lagrangian approach mainly refer to acceleration equations, i.e. differential equations of velocities. They are directly from the Euler-Lagrangian equations, and usually have higher-order terms truncated when they remain at the same post-Newtonian order of the Lagrangian. In this sense, they are incoherent equations of the Lagrangian and approximately conserve constants of motion in this system. In this paper, we show that the Euler-Lagrangian equations can also yield the equations of motion for consistency of the Lagrangian in the general case. The coherent equations are the differential equations of generalized momenta rather than those of the velocities, and have no terms truncated. The velocities are not integration variables, but they can be solved from the algebraic equations of the generalized momenta with an iterative method. Taking weak relativistic fields in the Solar System and strong relativistic fields of compact objects as examples, we numerically evaluate the accuracies of the constants of motion in the two sets of equations of motion. It is confirmed that these accuracies well satisfy the theoretical need if the chosen integrator can provide a high enough precision. The differences in the dynamical behavior of order and chaos between the two sets of equations are also compared. Unlike the incoherent post-Newtonian Lagrangian equations of motion, the coherent ones can theoretically, strictly conserve all integrals in some post-Newtonian Lagrangian problems, and therefore are worth recommending.

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Second post-Newtonian Lagrangian dynamics of spinning compact binaries

Cited by 40 publications

References 54 publications

On the conservation of the Jacobi integral in the post-Newtonian circular restricted three-body problem

On the conservation of the Jacobi integral in the post-Newtonian circular restricted three-body problem

Extended phase-space symplectic-like integrators for coherent post-Newtonian Euler-Lagrange equations

The equations of motion for consistency of a post-Newtonian Lagrangian formulation

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