哈密顿速端图在狭义和广义相对论中的命运
The fate of Hamilton's Hodograph in Special and General Relativity
摘要：
非相对论性粒子运动在欧几里得空间中的速端图是由其动量矢量描述的曲线。对于一般的中心轨道问题，速端图是轨道的踏板曲线的逆曲线（即其极倒数），旋转通过直角。汉密尔顿表明，开普勒/库仑问题，速端是一个圆的中心是在方向上的一个保守的偏心率向量。增加一个立方反比定律力会使偏心率矢量进动，从而使速度图进动。宇宙弦也会产生同样的效果。如果我们用相对论动量来定义速端线，那么对于索末菲（即特殊的相对论开普勒/库仑问题），存在一个有效的立方反力，它导致速端线进动。如果使用史瓦西坐标，也可以定义一个绕黑洞运动的类时测地线或零测地线的速端图。给出了它们的踏板方程。在特殊情况下，速端图可以显式地找到。例如，光子从过去奇点出发，掠过视界并返回未来奇点的轨道是心形线，其踏板方程是凯莱六次曲线，其逆是奇豪森三次曲线。本文还证明了在Beltrami坐标下，双曲空间上非相对论性Kepler问题的速度图也是圆.一个类似的结果适用于圆的3-球。在附录中，定义了在具有左不变度量的群流形上自由运动的质点的速端图。
英文摘要：
The hodograph of a non-relativistic particle motion in Euclidean space is the curve described by its momentum vector. For a general central orbit problem the hodograph is the inverse of the pedal curve of the orbit, (i.e. its polar reciprocal), rotated through a right angle. Hamilton showed that for the Kepler/Coulomb problem, the hodograph is a circle whose centre is in the direction of a conserved eccentricity vector. The addition of an inverse cube law force induces the eccentricity vector to precess and with it the hodograph. The same effect is produced by a cosmic string. If one takes the relativistic momentum to define the hodograph, then for the Sommerfeld (i.e. the special relativistic Kepler/Coulomb problem) there is an effective inverse cube force which causes the hodograph to precess. If one uses Schwarzschild coordinates one may also define a a hodograph for timelike or null geodesics moving around a black hole. Iheir pedal equations are given. In special cases the hodograph may be found explicitly. For example the orbit of a photon which starts from the past singularity, grazes the horizon and returns to future singularity is a cardioid, its pedal equation is Cayley's sextic the inverse of which is Tschirhausen's cubic. It is also shown that that provided one uses Beltrami coordinates, the hodograph for the non-relativistic Kepler problem on hyperbolic space is also a circle. An analogous result holds for the the round 3-sphere. In an appendix the hodograph of a particle freely moving on a group manifold equipped with a left-invariant metric is defined.
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