The Colombo top is a basic model in the rotation dynamics of a celestial body moving on a precessing orbit and perturbed by a gravitational torque. The paper presents a detailed study of analytical solution to this problem. By solving algebraic equations of degree 4, we provide the expressions for the extreme points of trajectories as functions of their energy. The location of stationary points (known as the Cassini states) is found as the function of the two parameters of the problem. Analytical solution in terms of the Weierstrass and the Jacobi elliptic functions is given for regular trajectories. Some trajectories are expressible through elementary functions: not only the homoclinic orbits, as expected, but also a special periodic solution whose energy is equal to that of the first Cassini state (unnoticed in previous studies).

中文翻译：

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科伦坡顶问题的解析解

科伦坡陀螺是天体在进动轨道上运动并受重力力矩扰动的旋转动力学的基本模型。本文详细研究了这个问题的解析解。通过求解 4 次代数方程，我们提供了轨迹极值点的表达式，作为其能量的函数。静止点的位置（称为卡西尼状态）被发现为问题的两个参数的函数。对于规则轨迹，给出了根据 Weierstrass 和 Jacobi 椭圆函数的解析解。一些轨迹可以通过初等函数来表达：不仅是同宿轨道，正如预期的那样，而且还有一个特殊的周期解，其能量等于第一个卡西尼态的能量（在以前的研究中没有注意到）。