
Page 166
By working the nbody problem in R, an increase in the number
of bodies will only lead to an increase of the value of the (activity) and
the (amount) of the particular integrals of Hamilton's equations.
Since the algebra of θfunctions is (for) a closed loop [7,35],
any disturbance of a Keplerian orbit will be expressed through elliptical
 functions, i.e., it is always defined by the (activity) of a certain
observed body of (disturbed) masses  bodies, that move (correspondent
with given).
In this way, the (activity) of the linear transformations, the invariance
of which satisfies the laws of Newtonian mechanics, transforms an integral
manifold of the nbody problem into a projective.
The result of this is a straightening of the trajectories for n
bodies, geodesics of a projective manifold, which is equivalent to solving
the problem in n1 elliptical quadratures. The result doesn't
depend on the correlations of masses, distances, and velocities at the
moment of time t_{0} in the framework of the initial formulation.
The question about the stability of this system  that should be the
subject for another article, but even now, we can notice that the integral
surface, according to Lie's results [8] is a minimal surface, and thus
the solution satisfies the principle of minimum action, i.e., it is the
only one.
In conclusion, the author counts it a pleasant obligation to express deep
gratitude to Victor Kuzmich Abalakin, Nikolai Viktorovich Dushin, Anatoly
Alexandrovich Efimov, Eduard Sergeevich Moskalev, Natalia Sergeevna
Petrova, Nikolai Nikolaevich Poliakov and Alexandra Alfredovna
Shpital'naya) for critical stimulation and discussion.
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[Some entries need tweaking.]
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