几何代数中的Kustaanheimo-Stiefel变换
The Kustaanheimo-Stiefel transformation in geometric algebra
摘要：
Kustaanheimo-Stiefel（KS）变换将三维开普勒问题的非线性奇异运动方程映射到四维谐振子的线性正则方程。它被广泛用于天体力学和原子物理学中的微扰开普勒问题的研究。与传统的基于矩阵的方法相比，在几何Clifford代数的语言中的KS变换的公式化提供了更清晰的几何解释和更大的计算简单性的优点。它表明，几何代数形式主义可以很容易地用于导出的拉格朗日和哈密顿描述的KS动力学在任意静态电磁场。对于从库仑中心开始的轨道，初始条件推导和框架的建立，允许讨论这些轨道的稳定性。
英文摘要：
The Kustaanheimo-Stiefel (KS) transformation maps the non-linear and singular equations of motion of the three-dimensional Kepler problem to the linear and regular equations of a four-dimensional harmonic oscillator. It is used extensively in studies of the perturbed Kepler problem in celestial mechanics and atomic physics. In contrast to the conventional matrix-based approach, the formulation of the KS transformation in the language of geometric Clifford algebra offers the advantages of a clearer geometrical interpretation and greater computational simplicity. It is demonstrated that the geometric algebra formalism can readily be used to derive a Lagrangian and Hamiltonian description of the KS dynamics in arbitrary static electromagnetic fields. For orbits starting at the Coulomb centre, initial conditions are derived and a framework is set up that allows a discussion of the stability of these orbits.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
