经典力学导论中的开普勒轨道与狭义相对论
Kepler's Orbits and Special Relativity in Introductory Classical Mechanics
摘要：
利用拉格朗日形式探讨了开普勒轨道的修正，由于狭义相对论。一个非常简单的模型只包括相对论动能，通过定义一个与狭义相对论和牛顿引力的相对论动量一致的拉格朗日量。在开普勒极限下求解相应的运动方程，得到了近似的相对论轨道方程，该方程与广义相对论在同一极限下导出的方程具有相同的形式，并清楚地描述了相对论开普勒轨道的三个特征：近日点的岁差、圆轨道半径的减小和偏心率的增大。对近日点进动速率的预测与仅用狭义相对论的已有计算结果一致。所有这三个特性在定性上都是正确的，尽管与更精确的广义相对论计算相比，它们被抑制了。这个模型是改进后，包括相对论引力势能。由此得到的近似相对论轨道方程与用非常简单的模型导出的方程具有相同的形式和对称性，并且更精确地描述了相对论轨道的特征。例如，水星近日点的岁差率的预测是广义相对论的三分之一。这些拉格朗日公式的特殊相对论开普勒问题是等价的熟悉的向量微积分公式。在开普勒极限下，这些模型应该是基于运动方程与使用广义相对论导出的运动方程的相似性的物理模型。这个轨道方程的推导对于物理专业的本科生和没有上过相对论课程的非专业人员来说是容易的。
英文摘要：
Kepler's orbits with corrections due to Special Relativity are explored using the Lagrangian formalism. A very simple model includes only relativistic kinetic energy by defining a Lagrangian that is consistent with both the relativistic momentum of Special Relativity and Newtonian gravity. The corresponding equations of motion are solved in a Keplerian limit, resulting in an approximate relativistic orbit equation that has the same form as that derived from General Relativity in the same limit and clearly describes three characteristics of relativistic Keplerian orbits: precession of perihelion; reduced radius of circular orbit; and increased eccentricity. The prediction for the rate of precession of perihelion is in agreement with established calculations using only Special Relativity. All three characteristics are qualitatively correct, though suppressed when compared to more accurate general-relativistic calculations. This model is improved upon by including relativistic gravitational potential energy. The resulting approximate relativistic orbit equation has the same form and symmetry as that derived using the very simple model, and more accurately describes characteristics of relativistic orbits. For example, the prediction for the rate of precession of perihelion of Mercury is one-third that derived from General Relativity. These Lagrangian formulations of the special-relativistic Kepler problem are equivalent to the familiar vector calculus formulations. In this Keplerian limit, these models are supposed to be physical based on the likeness of the equations of motion to those derived using General Relativity. The derivation of this orbit equation is approachable by undergraduate physics majors and nonspecialists whom have not had a course dedicated to relativity.
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