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On the Integration of the N-Body Equations

Part 3

Page 163 From (3.8) we obtain the second equation of the center of gravity for the first sub-system...
It follows, from the equations for the general center of gravity, that the
values of the sums
are independent of the (moment)
This (consists of) a body of mass
Page 164
are calculated semi-axes of Kepler's orbit μ bodies are moving. Orbits of bodies with masses
_{i}m_{1}, m_{2} are built as free
(multiplications) of n-1 Keplerian orbits.
In this way, the problem of n bodies with any
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.
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
,
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
[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)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
, 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.
Page 165
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
as (the) number of constants, which define the obtained solutions. So, with any
all 6 are solvable explicitly in quadrature. Let us look at isoparametric equations (see [42])
where
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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