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

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.

References in corpus (12)

• Inspiralling compact binaries in quasi-elliptical orbits: The complete third post-Newtonian energy flux
• Construction of Explicit Symplectic Integrators in General Relativity. I. Schwarzschild Black Holes
• Construction of explicit symplectic integrators in general relativity. II. Reissner-Nordstrom black holes
• Construction of explicit symplectic integrators in general relativity. IV. Kerr black holes
• Symplectic Integration of Post-Newtonian Equations of Motion with Spin
• Applying explicit symplectic integrator to study chaos of charged particles around magnetized Kerr black hole
• A Note on the Construction of Explicit Symplectic Integrators for Schwarzschild Spacetimes
• Integrability of eccentric, spinning black hole binaries up to second post-Newtonian order
• Explicit symplectic methods in black hole spacetimes
• Extended phase-space symplectic-like integrators for coherent post-Newtonian Euler-Lagrange equations
• Eccentric Motion of Spinning Compact Binaries
• An eccentric binary blackhole in post-Newtonian theory