AI 中文总结
研究固定能量下平面相对论性双中心问题，利用时间重参数化结合莫培督泛函的变分方法，构造无碰撞相对论轨迹并编码动力学，证明了有界轨道、特定散射解及捕获轨迹的存在。
AI 中文摘要
我们研究了固定能量下的平面相对论性双中心问题，包括一类开普勒势的微扰。通过时间重参数化，相对论动力学等同于具有临界强力奇点的经典双中心系统。这使我们能应用基于莫培督泛函的变分方法。我们构造了具有规定符号行程的无碰撞相对论轨迹，并通过可允许序列对动力学进行编码。特别地，我们证明了有界轨道、具有规定渐近方向的散射解以及在一个时间方向渐近自由且在另一个方向呈现规定符号行为的捕获轨迹的存在性。
英文摘要
We study the planar relativistic two-centre problem at fixed energy, including a class of perturbations of the Keplerian potential. Up to a reparametrisation of time, the relativistic dynamics is equivalent to a classical two-centre system with critical strong-force singularities. This reduction allows us to apply variational methods based on the Maupertuis functional. We construct collision-free relativistic trajectories with prescribed symbolic itineraries and obtain a coding of the dynamics by admissible sequences. In particular, we prove the existence of bounded orbits, scattering solutions with prescribed asymptotic directions, and trapped trajectories that are asymptotically free in one time direction and exhibit prescribed symbolic behaviour in the other.
Comments32 pages
