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On the Motion of a Three-Body System on Hypersurface of Proper Energy~1

机译：关于三体系统在适当能量〜1的超曲面上的运动

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Based on the fact that for hamiltonian system there exists equivalence between phase trajectories and geodesic trajectories on the Riemannian manifold, the classical three-body problem is formulated in the framework of six ordinary differential equations (ODEs) of the second order on the energy hypersurface of body system. It is shown that in the case when the total interaction potential of the body system depends on the relative distances between particles, the three of six geodesic equations describing rotations of formed by three bodies triangle are solved exactly. Using this fact, it is shown that the three-body problem can be described in the limits of three nonlinear ODEs of canonical form, which in phase space is equivalent to the autonomous sixth-order system. The equations of geodesic deviations on the manifold - (the space of relative distances between particles) are derived in an explicit form. A system of algebraic equations for finding the homographic solutions of restricted three-body problem is obtained. The initial and asymptotic conditions for solution of the classical scattering problem are found.
机译：基于哈密顿系统在黎曼流形上相轨迹和测地轨迹之间存在等价这一事实，经典三体问题是在能量超曲面的二阶六个常微分方程（ODE）的框架内提出的。身体系统。结果表明，在物体系统的总相互作用势取决于粒子之间的相对距离的情况下，描述由三个物体三角形形成的旋转的六个测地线方程中的三个精确解了。利用这一事实，表明可以在规范形式的三个非线性ODE的极限中描述三体问题，这在相空间中相当于自治六阶系统。流形上的测地线偏差方程（（粒子之间的相对距离的空间）以显式形式导出。获得了一个代数方程组，用于寻找约束三体问题的单应性解。找到了解决经典散射问题的初始条件和渐近条件。

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