Non-truncated strategy to exactly integrate the post-Newtonian Lagrangian circular restricted three-body problem

Huang, Li; Mei, Lijie; Shi-xiang, Huang

doi:10.1140/epjc/s10052-018-6291-1

Cited by 8 publications

(3 citation statements)

References 54 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Here, the integrator, dynamical parameters and initial orbital elements in the above Kepler problem are still used. In the present case, the speed of light c = 10 4 just corresponds to the first order post-Newtonian effect, which is approximate to the magnitude of 10 −8 (compared to the main Kepler part) in the solar system (Dubeibe et al 2017;Huang et al 2018). 4 As shown in Figure 3, the new method M1 and the Fukushima's method M2 give the machine double precision ǫ = 10 −16 to the inclination I and the longitude of ascending node Ω.…”

Section: A Post-newtonian Two-body Problemmentioning

confidence: 52%

The use of Kepler solver in numerical integrations of quasi-Keplerian orbits

Deng,

Wu,

Liang

2019

Preprint

View full text Add to dashboard Cite

A Kepler solver is an analytical method to solve a pure two-body problem. By slightly modifying the Kepler solver, we develop a new manifold correction method. Only one change to the analytical solution lies in that the obtainment of the eccentric anomaly does not need any iteration but uses the true anomaly between the constant Laplace-Runge-Lenz (LRL) vector and a varying radial vector given by an integrator. Such a new method rigorously conserves seven integrals of the Kepler energy, angular momentum vector and LRL vector. This leads to the numerical conservation of all orbital elements except the mean longitude. In the construction mechanism, the new method is unlike the existing Fukushima's linear transformation method that satisfies these properties. It can be extend to treat quasi-Keplerian orbits in perturbed two-body or N -body problems. The five slowly-varying orbital elements of each body relative to the central body are determined by the seven quasi-conserved quantities from their integral invariant relations, and the eccentric anomaly is calculated similarly in the above way. Substituting these values into the Kepler solver yields an adjusted solution of each body. Taking a post-Newtonian two-body problem and a six-body system of the Sun, four outer planets and Pluto as examples of quasi-Keplerian motions, we show that the new method can significantly improve the accuracies of all the orbital elements and the positions of individual planets, as compared with the case without correction. The new method and the Fukushima's method are almost the same in the numerical performance and need small additional computational cost.

show abstract

Section: A Post-newtonian Two-body Problemmentioning

confidence: 52%

The use of Kepler solver in numerical integrations of quasi-Keplerian orbits

Deng,

Wu,

Liang

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…In fact, ẍ0 and ÿ0 should be ẍ and ÿ, respectively. In this way, there are implicit acceleration equations suggested in [12]. They are given coherently, and should be completely equivalent to Eqs.…”

Section: Pn Circular Restricted Three-body Problemsmentioning

confidence: 99%

“…There is a question of whether the coherent Lagrangian equations of motion can be written. Recently, the au-thors of [12] suggested that coherent implicit acceleration equations, derived from the Lagrangian of a PN circular restricted three-body problem [4,13], should be integrated by using implicit numerical integrators. Unlike them, we provide a simple method to construct the coherent Lagrangian equations of motion that strictly conserve the constants of motion.…”

Section: Introductionmentioning

confidence: 99%

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

Li¹,

Wang²,

Deng³

et al. 2019

Preprint

View full text Add to dashboard Cite

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.

show abstract

Generalized flow-composed symplectic methods for post-Newtonian Hamiltonian systems

Huang,

Zeng,

Niu

et al. 2024

J. Cosmol. Astropart. Phys.

View full text Add to dashboard Cite

Due to the nonseparability of the post-Newtonian (PN) Hamiltonian systems of compact objects, the symplectic methods that admit the linear error growth and the near preservation of first integrals are always implicit as explicit symplectic methods have not been currently found for general nonseparable Hamiltonian systems. Since the PN Hamiltonian has a particular formulation that includes a dominant Newtonian part and a perturbation PN part, we present the generalized flow-composed Runge-Kutta (GFCRK) method with a free parameter λ to PN Hamiltonian systems. It is shown that the GFCRK method is symplectic once the underlying RK method is symplectic, and it is symmetric once the underlying RK method is symmetric under the setting λ = 1/2. Numerical experiments with the 2PN Hamiltonian of spinning compact binaries demonstrate the higher accuracy and efficiency of the symplectic GFCRK method than the underlying symplectic RK method in the case of weak PN effect. Meanwhile, the numerical results also support higher efficiency of the symplectic GFCRK method than the semi-explicit mixed symplectic method of the same order.

show abstract

Non-truncated strategy to exactly integrate the post-Newtonian Lagrangian circular restricted three-body problem

Cited by 8 publications

References 54 publications

The use of Kepler solver in numerical integrations of quasi-Keplerian orbits

The use of Kepler solver in numerical integrations of quasi-Keplerian orbits

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

Generalized flow-composed symplectic methods for post-Newtonian Hamiltonian systems

Contact Info

Product

Resources

About