用径向action坐标改进椭圆轨道的数值标绘--是否忽略了基于惯性对重力的莱布尼茨径向理论的对称性？
Improved numerical plotting of elliptical orbits using radial action coordinates -- has the symmetry of Leibniz radial theory based on inertia versus gravity been ignored?
摘要：
我们表明两体引力轨道可以用径向参考系而不是通常的牛顿直线惯性系来画。莱布尼茨是微积分学的创始人之一，也是牛顿在欧洲大陆的同时代人，他声称径向二阶导数可以通过计算半径立方的惯性力与半径平方反比的引力之差来求出。他的径向方法受到牛顿和支持者的严厉批评，他们更喜欢直线惯性系，还声称莱布尼茨未能满足牛顿的第三定律。我们表明，这两种方法是等价的，如果中心体是远远大于在轨道上。此外，我们证明了莱布尼茨最小作用的方法使用的半latus直角的轨道，特点的偏心率计算牛顿的向心加速度反对的惯性加速度，与净径向加速度控制半径也允许估计的角位移和行动的每一个间隔。数值上，我们的莱布尼茨模型可以更准确，因为它不需要假设线性向量充分模拟曲线轨迹。我们的牛顿引力方程使用质心坐标的审查的结论是，第三定律是满足对称性，如果引力和惯性质量分别采取的重力和径向惯性的等效力的表达式，直接到质心的径向惯性。这种对称性验证了质心径向坐标可以更准确地绘制包括双星在内的轨道，可能适用于更好地理解爱因斯坦相对论。径向坐标的一个优点是，轨道耦合也可以接受来自其他附近物体的向心扰动，但不是惯性。
英文摘要：
We that show two body gravitational orbits may be plotted using a radial reference frame rather than the customary Newtonian rectilinear inertial frame. Infinitesimal calculus cofounder and continental contemporary of Newton, Leibniz claimed that the second radial derivative could be found by taking the difference between an inertial force varying inversely, with the cubed radius and the gravitational force varying with an inversed squared radius. His radial method was severely criticised by Newton and supporters, who preferred the rectilinear inertial frame, also claiming Leibniz failed to satisfy Newton's third law of gravity. We show that these two approaches are equivalent if the central body is much larger than that in orbit. Furthermore we justify the Leibniz least action approach using the semi-latus rectum of the orbit, characteristic of the eccentricity to calculate the Newtonian centripetal acceleration opposed by the inertial acceleration, with the net radial acceleration controlling radius also allowing estimation of angular displacement and the action for each interval. Numerically, our Leibniz model can be more accurate since it requires no assumption that linear vectors adequately simulate a curvilinear trajectory. Our review of the Newton gravitational equation using centre of mass coordinates concludes that the third law is satisfied for symmetry if the gravitational and inertial masses are taken separately as equating force expressions for gravity and radial inertia, directed to the centre of mass for radial inertia. This symmetry validates centre of mass radial coordinates for more accurate plotting of orbits including dual stars, possibly applicable for better understanding of Einstein relativity. An advantage of radial coordinates is that orbiting couples can also accept perturbations from other nearby bodies acting centripetally, but not inertially.
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