Equations of motion derived from the Euler–Lagrange formula are usually implicit differential equations in terms of accelerations for the post-Newtonian (PN) Lagrangian dynamics of general relativistic problems. Meanwhile, the Legendre transform is in a nonclosed form due to the high nonlinearity of the velocity in the PN Lagrangian. To numerically integrate the exact implicit equations of motion rather than the truncated equations, the nontruncated strategy with an additional iteration for the Legendre transform is needed. In this paper, we aim at exploring energy-preserving methods accompanying the nontruncated strategy for the PN Lagrangian dynamics, once noticing that energy preservation as well as symplecticity is one of the most important properties of the Hamiltonian system that is completely equivalent to the PN Lagrangian. Incorporating the nontruncated strategy into the continuous-stage Runge–Kutta method, we thus propose a novel class of practical energy-preserving schemes. Numerical results from the PN Lagrangian circular restricted three-body problem and the spinning compact binaries show that the proposed numerical scheme is of high efficiency and can preserve energy nearly as accurately as machine precision.

从Euler–Lagrange公式得出的运动方程通常是广义相对论问题的后牛顿（PN）拉格朗日动力学加速度方面的隐式微分方程。同时，由于PN拉格朗日速度的高度非线性，勒让德变换为非封闭形式。为了在数值上积分精确的运动隐式方程而不是截断方程，需要为Legendre变换提供附加迭代的非截断策略。在本文中，我们旨在探索伴随PN拉格朗日动力学的非截断策略的节能方法，曾经注意到能量保存和辛辛度是哈密顿体系最重要的特性之一，它完全等同于PN拉格朗日。将非截断策略纳入连续阶段的Runge-Kutta方法，因此我们提出了一类新颖的实用节能方案。PN拉格朗日圆限制三体问题和旋转紧密二元方程的数值结果表明，所提出的数值方案具有很高的效率，并且几乎可以像机器精度一样精确地保存能量。