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#### Konstantin Manuilov On the Integration of the N-Body Equations Page 4

Page 166

By working the n-body 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 θ-functions 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 n-body 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 n-1 elliptical quadratures. The result doesn't depend on the correlations of masses, distances, and velocities at the moment of time t0 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.

#### Literature

[Some entries need tweaking.]

1. Newton, I., Mathematical Principia of Natural Philosophy, by Krivov, Vol. VII, Moscow, Inst. Acad. Sci. USSR, 1936
2. Manuilov, K.V., On the Integration of the N-Body Problem, in Development of Methods of Astronomical Research, Moscow-Leningrad, Acad. Sci. USSR, 1979, pp.300-325.
3. Apollonius from Pergamum, Eight Books of Courses (Conical Sections), Oxford, Vol. III, 1718 (Latin-Greek).
4. Abel, N.H., About the one general quality of a certain wide class of transcendental functions, Full collection of works, Vol. I, Christiania, Groudzel and Sons, 1882, pp. 145-212 (French).
5. Jacobi, K.G.J., New Foundations of the Theory of Ellipitical Functions (Latin), Coll. Works, Berlin, Rimer (German), 1881, pp. 482-504.
6. Riemann, B., The Theory of Abel's Functions, Moscow-Leningrad, G.I.T.T.L., 1948, p. 88 (German).
7. Riemann, B., On the Theory of Abel's Functions for Case P-3, Full Coll. Math. Works, Leipzig, Toebner, 11876, pp. 456-472 (German).
8. Lie, S. Full Coll. Works, Vol. II, Part 2, Oslo-Liepzig, Ashecog-Toebner. 1937 (German).
9. Poincaré, A., Different remarks about Abel functions theory, Works, Vol. 4, Paris, Gauthier-Villars, 1950, pp. 384-486 (French).
10. Aristotle, On the Heavens, Vol. 3. Moscow, "Misc.," 1981, pp. 263-278.

Page 167

11. Ptolomey, C., The Great Mathematical Construction, Vols. I, II, Leipzig, Toebner, 1898 (Greek).
12. Copernicus, N., Motions of the Heavenly Spheres, Full Coll. Works, Vol. II, Warsaw-Krakow, 1975 (Latin).
13. Kepler, J., New Astronomy (Latin). Full Coll. Works, Vol III, Munich, Kaspar, (German), 1937.
14. Vasiliev, H.P. and B.L. Gutenmacher, Straight Lines and Curves, Moscow, Science, pp. 122-123.
15. Euclid, Extended Coll. Works, Oxford, Vol. III, 1703 (Latin-Greek).
16. Archimedes, Writings, Moscow, GIFML, 1962.
17. Euler, L., New Theory of the Moon, Coll. Works, A.N. Krilov, Ed., Vols. 5-6, Moscow-Leningrad, Inst. Acad. Sci., USSR, 1937.
18. Euler, L. Mechanics, or the Science of Motion, Presented Analytically, Vols. I, II, Liepzig-Berlin, Toebner, 1912 (Latin).
19. Euler, L., The Investigation of the Three Body Problem, History of the West, Berlin Royal Academy of Science, Vol. 19, 1763, pp. 194-220 (French).
20. Bruns, H., About the Integration of the Problem of Several Bodies, Acta. Math., Vol. II, 1888, pp. 25-96 (German).
21. Poincaré, A., New Methods of Celestial Mechanics, Selected Works, Moscow, Science, 1972, Chap. 5.
22. (Penleve), P., About the Main Integrals of the Problem of N Bodies, Scientific Works, Vol. 3, Paris, 1975, pp. 666-698 (French).
23. Euler, L., Integral Calculations, Vol. I, Moscow, GITTL, 1956.
24. Euler, L., The Problem of Orthogonal Trajectories on a Surface, Roy. Acad. Sci., Vol. 7, St. Petersburg, 1820, pp. 33-60 (Latin).
25. Monge, G., Use of Analysis in Geometry, Moscow-Leningrad,ONII, 1958.
26. Euler, L., Integral Calculations, Vol. III, Moscow, GIFML, 1958.
27. Korkin, A.N. and E.I. Zolotorev, About Positively Defined Quadratic Forms, Coll. of Works of E.I. Zolotarev, Inst. Acad. Sci. USSR, Leningrad, 1931 (French).
28. Euler, L., The Theory of the Motion of Solid Bodies, Full Coll. Works, Vols. III and IV, Leipzig-Berlin, Toebner, 1948, 1950 (Latin).
29. Pontriagin, L.S., Continuous Groups, Moscow, Science, 1973.
30. Poincaré, A., New Methods of Celestial Mechanics, Selected Works, Vol. I, Moscow, Science, 1971, p. 20ff.
31. Arnold, V.I., Mathematical Methods of Classical Mechanics, Moscow, Science, 1976, p. 23, 60.
32. (Veil), G.K., On the Geometry of the Infinitely Small: Correspondence of Projective and Conformed Definitions, Full Coll. Works, Vol. 2, Berlin-Heidelberg-New York, 1968, pp. 195-207 (German).
33. Kagan, V.F., Sub-projective dimensions, Moscow, GIFML, 1961, p. 31.
34. Gromov, M.L., Topological Methods for Solutions of Differential Equations, Nice, 1970, Moscow, Science, 1972, pp. 72-76.
35. Klien, F., Non-Euclidian Geometry, Moscow, ONTI, 1936.
36. (Cobl), A., Algebraic Geometry and θ-Functions, KAMO, Vol. 10, New York, 1929 (English).
37. Beltrami, E., Foundations of the Theory of Dimensions of Constant Curvature, Mathematical Works, Vol. I, Milan, ILNB, 1905, pp. 186-228 (Italian).
38. Pontriagin, L.S., Fundamentals of Combinatorial Topology, Moscow, Science, 1976.

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