标题： N粒子相互作用系统平稳轨迹的构造 显示英文标题

标题： Construction of stationary trajectories for a model of a system of N particles with interaction

摘要： 对于经典的N体问题，提出了一种方法，即基于引入一些在物理意义上自然的数学规划优化问题，以在系统的可能状态集合上找到系统特性的条件极小值。 然后，这些优化问题的解使得能够构造系统的平面稳态和周期轨迹族，并且还可以找到这些轨迹上系统特性的关系和估计值。 研究表明，当系统在平面上沿着由这些优化问题的全局最小值生成的轨迹运动时，在每个当前“凝聚”（或势能）水平上，系统都能达到可能的最小尺寸。 类似地，还考虑了用于在三维空间中寻找系统特性条件极小值的优化问题。研究表明，这些问题的解只能在系统的平面轨迹上实现，并且特别地可以在构造出的平面稳态和周期轨迹上实现。 此外，还表明，在三维空间中的系统轨迹，至少在某一点上，当系统凝聚力（或势能）达到当前值时，系统能够达到可能的最小尺寸，这种轨迹只能是平面轨迹。而这些轨迹特别包括由所讨论的优化问题的全局最小值生成的平面稳态和周期轨迹。 显示英文摘要

摘要： For the classical N-body problem, an approach is proposed based on the introduction of some natural in the physical sense optimization problems of mathematical programming for finding a conditional minimum for the characteristics of the system on the set of its possible states. The solution of these problems then makes it possible to construct families of flat stationary and periodic trajectories of the system and also to find relationships and estimates for the characteristics of the system on these trajectories. It is shown that when the system moves on a plane on trajectories generated by the global minimum in these optimization problems, at any time the minimum possible size of the system is achieved at each current level of its "cohesion" (or potential energy). Similar optimization problems are considered for finding a conditional minimum for the characteristics of a system in three-dimensional space. It is shown that the solution of these problems can be achieved only on flat trajectories of the system and is achieved, in particular, on the constructed flat stationary and periodic trajectories. In addition, it is shown that the trajectory of the system in three-dimensional space, at least at one point of which the minimum possible size of the system is achieved at the current value of its cohesion (or potential energy), can only be flat. And such trajectories are, in particular, flat stationary and periodic trajectories generated by the global minimum in the considered optimization problems.