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

TL;DR

This paper compares the implicit method in incoherent Lagrangian systems with the correction map method in Hamiltonian systems within post-Newtonian gravity, demonstrating the correction map's superior ability to conserve motion constants.

Contribution

It introduces a comparison between two numerical methods in post-Newtonian mechanics, highlighting the effectiveness of the correction map method over the implicit Lagrangian approach.

Findings

Correction map method better preserves angular momentum.

Numerical simulations show improved invariance of constants.

Correction map outperforms implicit algorithms in accuracy.

Abstract

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 Lagrangian to 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.