The Kepler–Heisenberg problem is that of determining the motion of a planet around a sun in the Heisenberg group, thought of as a three-dimensional sub-Riemannian manifold. The sub-Riemannian Hamiltonian provides the kinetic energy, and the gravitational potential is given by the fundamental solution to the sub-Laplacian. The dynamics are at least partially integrable, possessing two first integrals as well as a dilational momentum which is conserved by orbits with zero energy. The system is known to admit closed orbits of any rational rotation number, which all lie within the fundamental zero-energy integrable subsystem. Here, we demonstrate that, under mild conditions, zero-energy orbits are self-similar. Consequently, we find that these zero-energy orbits stratify into three families: future collision, past collision, and quasi-periodicity without collision. If a collision occurs, it occurs in finite time.

开普勒-海森堡问题是确定海森堡组中行星围绕太阳运动的问题，该组被认为是三维次黎曼流形。子黎曼哈密顿函数提供动能，而引力势由子拉普拉斯定理的基本解给出。动力学至少是部分可积分的，具有两个第一积分以及一个扩张动量，该动量由零能量的轨道守恒。已知该系统允许任何有理旋转数的闭合轨道，所有闭合轨道都位于基本的零能量可积分子系统之内。在这里，我们证明了在温和条件下零能量轨道是自相似的。因此，我们发现这些零能量轨道分为三个族：未来碰撞，过去碰撞，和准周期而无碰撞。如果发生碰撞，则会在有限时间内发生。