开普勒积分的非保守扩张与一类新的可积系统
Nonconservative extension of Keplerian integrals and a new class of integrable system
摘要：
在开普勒问题中，拉格朗日量在时间平移和旋转下的不变性产生了与能量和角动量有关的守恒定律。诺特定理揭示了这些对称性在特殊的非保守情形下提供了第一积分的广义形式，它近似于各种物理模型。该系统是由双参数加速度与组件沿切向和法向方向的扰动。一个相似变换减少了双参数的干扰，一个简单的单参数迫使沿速度矢量。讨论了这一新问题的可解性条件，并给出了可积情形的封闭解。由于广义能量守恒，轨道被分类为椭圆，抛物线和双曲。开普勒轨道很自然地表现为这个问题的特殊解。在对轨道进行独立刻画后，基于Weierstrass椭圆函数建立了解的统一形式。新的轨迹涉及基本曲线，如心形线和对数，正弦曲线和Cotes'螺旋。这些轨道可以代表受太阳辐射压力扰动的粒子的运动，具有连续推力推进的航天器的运动，以及史瓦西测地线的一些实例。最后，将这个问题与天体力学中其它已知的可积系统联系起来。
英文摘要：
The invariance of the Lagrangian under time translations and rotations in Kepler's problem yields the conservation laws related to the energy and angular momentum. Noether's theorem reveals that these same symmetries furnish generalized forms of the first integrals in a special nonconservative case, which approximates various physical models. The system is perturbed by a biparametric acceleration with components along the tangential and normal directions. A similarity transformation reduces the biparametric disturbance to a simpler uniparametric forcing along the velocity vector. The solvability conditions of this new problem are discussed, and closed-form solutions for the integrable cases are provided. Thanks to the conservation of a generalized energy, the orbits are classified as elliptic, parabolic, and hyperbolic. Keplerian orbits appear naturally as particular solutions to the problem. After characterizing the orbits independently, a unified form of the solution is built based on the Weierstrass elliptic functions. The new trajectories involve fundamental curves such as cardioids and logarithmic, sinusoidal, and Cotes' spirals. These orbits can represent the motion of particles perturbed by solar radiation pressure, of spacecraft with continuous thrust propulsion, and some instances of Schwarzschild geodesics. Finally, the problem is connected with other known integrable systems in celestial mechanics.
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