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The equations of motion for a Lagrangian mainly refer to the accelerationequations, which can be obtained by the Euler--Lagrange equations. In thepost-Newtonian Lagrangian form of general relativity, the Lagrangian systemscan only maintain a certain post-Newtonian order and are incoherent Lagrangianssince the higher-order terms are omitted. This truncation can cause somechanges in the constant of motion. However, in celestial mechanics,Hamiltonians are more commonly used than Lagrangians. The conversion fromLagrangian to Hamiltonian can be achieved through the Legendre transformation.The coordinate momentum separable Hamiltonian can be computed by the symplecticalgorithm, whereas the inseparable Hamiltonian can be used to compute theevolution of motion by the phase-space expansion method. Our recent workinvolves the design of a multi-factor correction map for the phase-spaceexpansion method, known as the correction map method. In this paper, we comparethe performance of the implicit algorithm in post-Newtonian Lagrangians and thecorrection map method in post-Newtonian Hamiltonians. Specifically, weinvestigate the extent to which both methods can uphold invariance of themotion's constants, such as energy conservation and angular momentumpreservation. Ultimately, the results of numerical simulations demonstrate thesuperior performance of the correction map method, particularly with respect toangular momentum conservation.

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