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Konstantin Manuilov
On the Integration of the N-Body Equations
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      From (3.8) we obtain the second equation of the center of gravity for the first sub-system...

many 3-11

      It follows, from the equations for the general center of gravity, that the values of the sums many 3-11a are independent of the (moment) t0 at which we are observing the system.
      Equations of the center of gravity of Newton's second sub-system can be obtained from the second components of the quadratic forms (3.7) and (3.8)

many 3-12

      This (consists of) a body of mass m2 and a general center of gravity for n-2 bodies. The line of action of forces of the second sub-system is othogonal to the line of the first, and the resultant of forces of the interaction of a body of mass m1 with bodies m2 and many 3-12a is applied exactly (to) the general center of gravity of the latter.
      If the system is for three bodies, then the reduction is complete. Transformation of (forms) (3.7, 3.8) to the main axes releases pairs of bodies from acting on one another (Theorem XXIV). From the equations for the centers of gravity of Newton's sub-systems

many 3-13

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are calculated semi-axes of Kepler's orbit aibi, on which μi bodies are moving. Orbits of bodies with masses m1, m2 are built as free (multiplications) of n-1 Keplerian orbits.
      In this way, the problem of n bodies with any many 3-13a is integrated in an elliptical coordinate system.
      For the definition of the orbit for any body of that system, it is necessary and sufficient to look at n/2 (n is even) or (n+1)/2 (n is odd) expressions of quadratic form, by changing (numerization) of the bodies by cyclic (permutations) (by twos).
      The equations of the orbits are represented by Abel's θ-functions for n-1 variables, which are, because of the diagonality of the quadratic forms, free (multiplications) of n-1 Jacobi's ellipical functions.

many 3-14

      The problem of three bodies in a straight line was examined first by Euler [19] and then by Lagrange [38], and is characterized by coefficients many 3-14a , that give the (transformation) of a system of three bodies into two free sub-systems, (which) are represented as non-real (one line of forces (acting)). Lagrange's system of three bodies with many 3-14b [39] during (diagonalization) of the quadratic forms are divided (into) free bodies which don't act on one another, the same, as with similar systems of four and six bodies.

4. Comparison of obtained results with (classical)

      Let us compare between obtained results and (classical). The solution of the n-body problem should satisfy the theorem of Bruns-Poincaré, i.e., (it) should be defined by 10 general integrals of motion - six coordinates for the centers of gravity, three Keplers and an integral of energy or (activity) S. Of these 10 integrals, six are linearly independent. All other integrals - for example, the 18 particular integrals of the three-body problem - with many 3-14c, should be linear (transformations) of these six.
      Let us do once again the separation of n-1 minima of quadratic form (3.8) on the axis of Descartes coordinates, according to Jacobi. Each of the component-minima, in expression (2.3), is related to the periods of the θ-functions.

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      Their free multiplication defines a structure of general and of particular solutions of the problem. In this way a number of constants, shown by the Bruns-Poincaré theorem, is exactly the same with any many 3-13a as (the) number of constants, which define the obtained solutions. So, with any many 3-13a all 6n coordinates of the centers of gravity, 3n of Kepler's and n in particular integrals of energy, become linear transformations (of) one from ten general, and the n-body problem is (for) any many 3-13a, a problem of 6th order (look at [19]). These particular integrals are obtained as a result of bringing the quadratic (forms) (activity) toward the main axes and represent, by themselves, solutions of canonical equations, which totally correspond to the Hamilton-Jacobi theorem [31].
      If the Hamilton-Jocobi solution S(Qq) is already known, and that solution is dependent on n parameters and is such that

many 4-00
then the canonical equations
many 4-01
are solvable explicitly in quadrature.
      Let us look at isoparametric equations (see [42])

many 4-02

many 4-2a
which define the harmony of physical and geometrical spaces on one, two or three dimensions. (The) dimension(s) of (the) absolutely invariant in R3 - (a) determinant in quadratic form, corresponds to the space(s) of (Liouville) [31], which (have) the dimensions of a cube of (activity). According to this, we, naturally, can suppose that a2 is proportional to the (activity) of the system.

many 4-3

many 4-4

      Let us notice that an increase in the dimensions of space, for example, a change in phase [31], will cause the creation of integrals of motion that do not appear in any of the three classes that are given by the Bruns-Poincaré theorem, as well as changing of the physical dimensions of the integrals of motion,and this will cause the (condition) in which it will be impossible to integrate the problem.

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