A Note on the Equivalence of Post-Newtonian Lagrangian and Hamiltonian Formulations

Chen, Rong-Chao; Wu, Xiaoxia

doi:10.1088/0253-6102/65/3/321

Cited by 16 publications

(9 citation statements)

References 41 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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“…3 Fortunately, another important result of [30] is that for a lower-order PN Lagrangian formulation with Euler-Lagrange equations to an infinite PN order there always exists a formally equivalent PN Hamiltonian at the infinite order from a theoretical point of view or a certain finite order from a numerical point of view. This result is supported analytically and numerically via a special 1PN Lagrangian formulation of a relativistic circular restricted three-body problem with the Euler-Lagrange equations and the equivalent Hamiltonian as a converging Taylor series [36]. Thus, the integrability or non-integrability of a PN Lagrangian formalism can be shown in terms of that of its formal equivalent PN Hamiltonian.…”

Section: Introductionmentioning

confidence: 73%

Second post-Newtonian Lagrangian dynamics of spinning compact binaries

Huang

2016

Eur. Phys. J. C

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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 leads to a 3PN spin-spin coupling occurring 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 this claim.

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“…This strategy is extensively used in the PN Lagrangian of compact binaries [8][9][10][11][12][13], the equivalence analysis between the PN Hamiltonian and the PN Lagrangian approaches at the same order [14][15][16][17][18][19][20], and the dynamics of the PN circular restricted three-body problem [1,5,6]. Following this idea, the approximately truncated motion equations at the 1PN order for (13) are derived in [5] and expressed as…”

Section: Exact and Truncated Motion Equationsmentioning

confidence: 99%

“…During this procedure, truncation is a commonly used approach, which is conducted by substituting low-order equations of motion to high-order accelerations or by neglecting higher-order terms. This strategy is extensively used to investigate the dynamics of the PN Lagrangian compact binaries [8][9][10][11][12][13], the equivalence analysis between the PN Lagrangian and the PN Hamiltonian approaches [14][15][16][17][18][19][20], and the dynamics of the PN circular restricted three-body problem [1,5,6]. Among the discussion of the equivalence between the PN Lagrangian and the PN Hamiltonian approaches, the authors of [14][15][16]19] showed the physical equivalence of the two approaches at the same PN order.…”

Section: Introductionmentioning

confidence: 99%

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

2018

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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 O(10 −12 ), whereas the truncated approach at the first post-Newtonian (1PN) order only has an accuracy of 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.

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A Note on the Equivalence of Post-Newtonian Lagrangian and Hamiltonian Formulations

Cited by 16 publications

References 41 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

Second post-Newtonian Lagrangian dynamics of spinning compact binaries

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

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