Coherent post-Newtonian Lagrangian equations of motion

Li, Dan; Wang, Yu; Deng, Chen; Wu, Xiaoxia

doi:10.1140/epjp/s13360-020-00407-7

Cited by 17 publications

(11 citation statements)

References 19 publications

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“…The equation and Eqs. ( 3) and (11) were called as the coherent or exact PN Lagrangian (or Euler-Lagrange) equations of motion in [29,30]. Naturally, the coherent equations have the consistency of energy E or Hamiltonian H. If p in Eq.…”

Section: Extended Phase-space Symplectic-like Integratorsmentioning

confidence: 99%

“…They numerically showed that the exact Euler-Lagrange equations always well conserve the Jacobi integral regardless of the choice of a and c. The Hamiltonian formalism can also strictly conserve the integrals of motion from the theory. Dubeibe et al [31] numerically confirmed that the conservation of the Jacobi integral in the Hamiltonian is independent of the choice of a and c. In a word, the approximate Euler-Lagrange equations, exact Euler-Lagrange equations and Hamiltonian equations at same PN orders have explicit differences for a strong gravitational field of compact objects, whereas have negligible differences for a weak gravitational field like the solar system [26,29,30,32]. The approximate Euler-Lagrange equations for a given PN Lagrangian formalism of compact objects are a set of wrong equations of motion and poorly conserve the constants of motion.…”

Section: Introductionmentioning

confidence: 96%

“…The other path is the differential equations of positions and generalized momenta, where velocities are solved from the algebraic equations of the generalized momenta via an iterative method. Such equations of motion are coherent (or exact) PN Lagrangian equations of motion [29,30]. The approximate Lagrangian equations approximately conserve the integrals of motion (e.g., energy) in the PN Lagrangian formalism in general, but the exact ones always strictly conserve the integrals of motion from a theoretical point of view.…”

Section: Introductionmentioning

confidence: 99%

“…Dubeibe et al [31] investigated the conservation of the Jacobi integral in a PN circular restricted three-body problem, and found that the approximate Euler-Lagrange equations do not not conserve the Jacobi integral for the distance a of the two primaries and the speed of light c satisfying the conditions a = c = 1, but conserve well for the case of a = 1 and c = 10 4 . The authors of [29,30] pointed out that the approximate Lagrangian equations and the exact ones do not have typical differences for the case of a = 1 and c = 10 4 , but have for the case of a = c = 1. They numerically showed that the exact Euler-Lagrange equations always well conserve the Jacobi integral regardless of the choice of a and c. The Hamiltonian formalism can also strictly conserve the integrals of motion from the theory.…”

Section: Introductionmentioning

confidence: 99%

“…The main aim of the present paper is to discuss a possible application of extended phase space symplectic-like integrators to the coherent PN Euler-Lagrange equations [29,30]. For this purpose, we construct symplectic-like integrators for the exact PN Euler-Lagrange equations in an extended phase space of a PN Lagrangian formalism in Sect.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Extended Phase-space Symplectic-like Integratorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Pan,

Wu,

Liang

2021

Preprint

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Beyond Newtonian Dynamics of Planar CRTBP with Kerr—Like Primaries

Roychowdhury,

Banerjee

2023

Lecture Notes in Physics

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In this article, we shall discuss the dynamics of the classic planar circular restricted three-body problem (CRTBP) with spinning primaries in the context of a beyond-Newtonian approximation [27]. We shall begin by discussing the construction of the beyond-Newtonian potential in the so-called Fodor–Hoenselaers–Perjés procedure where we keep first order non-Newtonian contributions in both the mass and spin. Using this potential, we shall then discuss our model for a test particle of infinitesimal mass orbiting in the equatorial plane of the two primaries. The talk shall then discuss the dynamics as the system transitions from the Newtonian to the beyond-Newtonian regime. We shall then study the evolution and stability of the fixed points of the system as a function of the parameter $$\epsilon $$ ϵ with the dynamics of the particle analyzed using the Poincaré map of section and the Maximal Lyapunov Exponent as indicators of chaos. We shall establish that the intermediate values of $$\epsilon $$ ϵ seem to be the most chaotic for the two cases of primary mass ratios $$(=0.001, 0.5)$$ ( = 0.001 , 0.5 ) examined. We also conclude that the amount of chaos in the system remains higher than the Newtonian system as well as for the planar circular restricted three-body problem with Schwarzschild-like primaries for all non-zero values of $$\epsilon $$ ϵ .

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Dynamics of charged particles around a magnetically deformed Schwarzschild black hole

Miao

2020

Phys. Scr.

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The motion of a charged particle around a deformed Schwarzschild black hole in an external magnetic field is studied. If a magnetic parameter is small, the sign and magnitude of deformation parameter affect the shape of effective potential. The radius of an innermost stable circular orbit on the equatorial plane increases as the magnitude of negative deformation parameter increases, but decreases as a positive deformation parameter increases. When the energy and angular momentum are given appropriately, the magnetic parameter has two critical values, small and large values. For the magnetic parameter smaller than the small value or larger than the large value, the deformation parameter does not have an explicit effect on the dynamics of particles on the non-equatorial plane. The former dynamics is ordered, whereas the latter dynamics is strongly chaotic. If the magnetic parameter runs from the small value to the large one, the negative deformation parameter is helpful to strengthen the extent of chaos with its magnitude increasing. However, chaos is weakened with an increase of the positive deformation parameter.

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Coherent post-Newtonian Lagrangian equations of motion

Cited by 17 publications

References 19 publications

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

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

Beyond Newtonian Dynamics of Planar CRTBP with Kerr—Like Primaries

Dynamics of charged particles around a magnetically deformed Schwarzschild black hole

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