post-Newtonian Lagrangian Research Articles

Contrasting the Implicit Method in Incoherent Lagrangian and the Correction Map Method in Hamiltonian

The equations of motion for a Lagrangian mainly refer to the acceleration equations, which can be obtained by the Euler–Lagrange equations. In the post-Newtonian Lagrangian form of general relativity, the Lagrangian systems can only maintain a certain post-Newtonian order and are incoherent Lagrangians since the higher-order terms are omitted. This truncation can cause some changes in the constant of motion. However, in celestial mechanics, Hamiltonians are more commonly used than Lagrangians. The conversion from Lagrangianto Hamiltonian can be achieved through the Legendre transformation. The coordinate momentum separable Hamiltonian can be computed by the symplectic algorithm, whereas the inseparable Hamiltonian can be used to compute the evolution of motion by the phase-space expansion method. Our recent work involves the design of a multi-factor correction map for the phase-space expansion method, known as the correction map method. In this paper, we compare the performance of the implicit algorithm in post-Newtonian Lagrangians and the correction map method in post-Newtonian Hamiltonians. Specifically, we investigate the extent to which both methods can uphold invariance of the motion’s constants, such as energy conservation and angular momentum preservation. Ultimately, the results of numerical simulations demonstrate the superior performance of the correction map method, particularly with respect to angular momentum conservation.

A Comparison between Second-Order Post-Newtonian Hamiltonian and Coherent Post-Newtonian Lagrangian in Spinning Compact Binaries

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.

Post-Newtonian Lagrangian of an N -body system with arbitrary mass and spin multipoles

The present paper derives the post-Newtonian Lagrangian of translational motion of N arbitrary-structured bodies with all mass and spin multipoles in a scalar-tensor theory of gravity. The multipoles depend on time and evolve in accordance with their own dynamic equations of motion. The Lagrangian is retrieved from the post-Newtonian equations of motion by solving the inverse problem of the Lagrangian mechanics and generalizes a well-known Lagrangian of pole-dipole-quadrupole massive particles to the particles of higher multipolarity. Analytic treatment of the higher-order multipole contributions is important for more rigorous computation of gravitational waveform of inspiralling compact binaries at the latest stage of their orbital evolution before merger when tidal and rotational deformations of stars are no longer small and rapidly change in time. The Lagrangian of an N-body system with arbitrary mass and spin multipoles is instrumental for formulation of the post-Newtonian conservation laws of energy, momenta and the integrals of the center of mass.

Symplectic integrators for post-Newtonian Lagrangian dynamics

In general, a truncation for the exact implicit equations of motion is necessary when quantitative research is carried out for the post-Newtonian Lagrangian dynamics of high nonlinearity. However, this truncation for equations of motion and the using of featureless numerical methods may lead to the essential gap between numerical and theoretical results because chaos amplifies the initial truncation error to an unacceptable level after long-term numerical integration. To provide highly accurate numerical solutions for post-Newtonian Lagrangian systems of compact objects, we investigate symplectic integrators incorporating the nontruncated strategy in this paper. By virtue of the canonical form of Hamiltonian dynamics, we propose an effective approach to construct symplectic integrators for the post-Newtonian Lagrangian dynamics, where an iteration procedure to obtain velocities from the generalized momentums is needed. That is, the post-Newtonian Lagrangian is numerically solved not in the Lagrangian frame but in the Hamiltonian frame. Numerical experiments with the post-Newtonian Lagrangian circular restricted three-body problem, and the Lagrangian spinning compact binaries at the second post-Newtonian order soundly support the efficiency and robustness of the proposed symplectic integrators. Finally, the mixed symplectic integrators are preferred because of their improved performance when the separation of one solvable dominate part with another perturbation part is possible for the equivalent Hamiltonian.

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

In this study, we present a novel non-truncated strategy by accompanying the fixed-point iteration with traditional numerical integrators. The proposed non-truncated strategy aims to exactly integrate implicit motion equations that are directly derived from the Lagrangian of the post-Newtonian circular restricted three-body problem. In comparison with the commonly used truncated approach, which cannot exactly but approximately preserve the generalized Jacobian constant (or energy) of the original Lagrangian system, the proposed non-truncated strategy has been determined to preserve this constant well. In fact, the non-truncated strategy and the truncated approach have a difference at second post-Newtonian order. Based on Kolmogorov–Arnold–Moser theory, this difference from the truncation in the equations of motion may lead to destroying the orbital configuration, dynamical behavior of order and chaos, and conservation of the post-Newtonian circular restricted three-body problem. The non-truncated strategy proposed in this study can avoid all these drawbacks and provide highly reliable and accurate numerical solutions for the post-Newtonian Lagrangian dynamics. Finally, numerical results show that the non-truncated strategy can preserve the generalized Jacobian constant in the accuracy of {mathscr {O}}(10^{-12}), whereas the truncated approach at the first post-Newtonian (1PN) order only has an accuracy of {mathscr {O}}(10^{-3}). Moreover, several orbits are observed to be escaping from the bounded region in the 1PN truncated system via the truncated strategy, but these escaping orbits are unobserved via the non-truncated strategy.

Second post-Newtonian Lagrangian dynamics of spinning compact binaries

The leading-order spin-orbit coupling is included in a post-Newtonian Lagrangian formulation of spinning compact binaries, which consists of the Newtonian term, first post-Newtonian (1PN) and 2PN non-spin terms and 2PN spin-spin coupling. This makes a 3PN spin-spin coupling occur in the derived Hamiltonian. The spin-spin couplings are mainly responsible for chaos in the Hamiltonians. However, the 3PN spin-spin Hamiltonian is small and has different signs, compared with the 2PN spin-spin Hamiltonian equivalent to the 2PN spin-spin Lagrangian. As a result, the probability of the occurrence of chaos in the Lagrangian formulation without the spin-orbit coupling is larger than that in the Lagrangian formulation with the spin-orbit coupling. Numerical evidences support the claim.

A Note on the Equivalence of Post-Newtonian Lagrangian and Hamiltonian Formulations**Supported by the the Natural Science Foundation of Jiangxi Province under Grant No. [2015]75 and the National Natural Science Foundation of China under Grant Nos. 11173012, 11178002, and 11533004

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.

Second-order parametrized-post-Newtonian Lagrangian.

A many-body Lagrangian to second post-Newtonian order using an extension of the parametrized-post-Newtonian (PPN) formalism is introduced and the properties of new parameters are explored. A parametrized gauge transformation is developed to permit comparison with theories of gravity in a variety of different coordinate systems. A procedure to impose Lorentz invariance on a general second-order post-Newtonian Lagrangian is developed. The Lagrangian is then constrained to possess Lorentz invariance and a "Lorentz-invariant" gauge is introduced. The constrained Lagrangian is found to be described by ten new second-order PPN parameters. When the Lagrangian is further constrained to describe theories of gravity for which test particles move along geodesics, one of the ten new parameters is given entirely in terms of first-order PPN parameters, leaving only nine PPN parameters to describe the second-order gravitational interaction. A "metric" gauge is introduced which allows the metric to be easily found from the Lagrangian and is shown to reduce to the gauge associated with the canonical formalism of Arnowitt, Deser, and Misner when the general-relativity values of the PPN parameters are used.

A many-body Lagrangian for celestial body dynamics to second post-Newtonian linear field order

view Abstract Citations (9) References (5) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS A Many-Body Lagrangian for Celestial Body Dynamics to Second Post-Newtonian Linear Field Order Benacquista, M. ; Nordtvedt, K. Abstract The many-body Lagrangian formalism developed by Nordvedt is used to develop an extension of the first post-Newtonian Lagrangian for compact celestial bodies interacting via an underlying metric theory of gravity. The Lagrangian is extended to second order in the motion of the bodies but is linear in the potentials. A hierarchy of Lorentz invariances is then imposed on this "linear field" Lagrangian, and the resulting Lorentz-invariant Lagrangian is found to depend solely on the structure of the first post-Newtonian Lagrangian. Publication: The Astrophysical Journal Pub Date: May 1988 DOI: 10.1086/166316 Bibcode: 1988ApJ...328..588B Keywords: Celestial Bodies; Euler Equations Of Motion; Gravitation Theory; Lorentz Force; Many Body Problem; Computational Astrophysics; Degrees Of Freedom; Astrophysics; GRAVITATION; RELATIVITY full text sources ADS |

Gravitational two-body problem with acceleration-dependent spin terms

We generalize our previous work, on the gravitational two-body post-Newtonian Lagrangian with spin and parametrized post-Newtonian parameters γ and β, by addingaccelerationdependent spin terms corresponding to anarbitrary spin supplementary condition. For the purpose of constructing the corresponding Hamiltonian we make use of our recently developedmethod of the double zero. Using this method, we remove the acceleration-dependent spin terms from the Lagrangian and, in the process, create new non-accelerationdependent terms. Use of this new Lagrangian enables us to construct the Hamiltonian corresponding to the arbitrary spin supplementary condition. Energy constants of the motion are also discussed.

Center of inertia and coordinate transformations in the post-Newtonian charged n-body problem in gravitation

We generalize the field theory propagator by finding a way to make it a function of some additional arbitrary parameters. Thus, it is now possible to obtain Lagrangians (which contain the propagator parameters) from field theory in a more general coordinate system than had previously been possible. We find the n-body (classical) Bażański Lagrangian in this more general coordinate system and we give the relationship between the various coordinate systems by an n-body coordinate transformation involving the propagator parameters. We find the center of inertia for the case of the n-body Basżański Lagrangian in the general coordinate system and find that the potential energy terms −Gmimj/rij and eiej/rij do not in general split equally between particles i and j as they do in the case of Bażański coordinates. We also find the center of inertia for the case of the n-body (unchanged) post-Newtonian Lagrangian with parameterized post-Newtonian (PPN) parameters γ and β in standard coordinates, and show that the potential energy terms do split equally between a pair of particles.

Post-Newtonian two-body and n-body problems with electric charge in general relativity

Starting with the Bażański two-body post-Newtonian Lagrangian with electric charge in general relativity, we construct a coordinate transformation (not involving center-of-mass coordinates) with two arbitrary parameters and obtain a Hamiltonian which is in agreement with one derived from quantum field theory. The field theory Hamiltonian corresponds to using an arbitrary parameter xp in the photon propagator as well as an arbitrary parameter xg in the graviton propagator. These results are also generalized to the case of n bodies. The condition for static balance ei=±G1/2mi is found to hold both for the exact Reissner–Nordstro/m ’’one-body’’ problem and for the post-Newtonian n-body problem. An alternate condition for static balance ei=± (Gm1m2)1/2 is found to hold for the post-Newtonian two-body problem. The precession of the perihelion for the post-Newtonian two-body problem is given along with four special cases, one of which is the two-body generalization of the ’’one-body’’ special relativity result of Sommerfeld. Post-Newtonian two-body equations of motion (in center-of-mass coordinates) with the condition of static balance are also examined.