  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.
