  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.
