Semen Sadetov

ul. Zorge, 7, 344090, Rostov, Russian Federation
Rostov State University, Rostov-on-Don


Sadetov S. T.
It is established that the restricted circular planar three-body problem (RCPTBP) [1], [15], [5] admits a nonconstant algebraic integral on a level of energy only in cases when it can be reduced to the Kepler problem. The Hill problem [1], [7], [5] is the limit case of the RCPTBP if by analogy with the Moon-Earth-Sun system we put the mass of the Sun and the distance between the Sun and the Earth to be infinitely large. It is established that the Hill problem also does not admit a non-constant algebraic integral on any level of energy. The proof is based on the Husson method [8], [2], improved by the author [21], [22]. At the end of the proof we expand the result of J. Liouville [13] that the integral $\int f(z)e^z$ for $f$ algebraic in $z$ is not generally an algebraic function times the exponent function.
Keywords: Hill problem, restricted circular planar three-body problem, algebraic integrals, improved Husson method, Hamiltonian perturbation
Citation: Sadetov S. T.,  On algebraic integrals of the Hill problem and restricted circular planar three-body problem on a level of energy , Regular and Chaotic Dynamics, 2005, vol. 10, no. 3, pp. 323-332
DOI: 10.1070/RD2005v010n03ABEH000318
Sadetov S. T.
After the lowering of the order up to the location of the center of inertia the reduction of this paper is performed by the passage to the complete set of invariants of the action of linear group of rotations and reflections on the phase space $\mathrm{T}^{*}\mathbb{R}^n \otimes \mathbb{R}^N$. In distinction to known reductions, this reduction is homeomorphic. For $N + 1 = 3,\, n=3,2$ the orbits of the coadjoint representation of the group $Sp(4)$, on which real motions take place, have the homotopy type of projective space $\mathbb{R}P^3$, sphere $S^2$, homogeneous space $(S^2\times S^1)/\mathbb{Z}^2$.
Citation: Sadetov S. T.,  On the Regular Reduction of the $n$-Dimensional Problem of $N+1$ Bodies to Euler–Poincaré Equations on the Lie Algebra $sp(2N)$, Regular and Chaotic Dynamics, 2002, vol. 7, no. 3, pp. 337-350
Sadetov S. T.
The motion of a point over an $n$-dimensional nondegenerate quadric in one-dimentioned quadratic potential under the assumption that there exist $n+1$ mutually orthogonal planes of symmetry is considered. It is established, that all cases of the existence of an algebraic complete commutative set of integrals are exhausted by classical ones. The question whether the integrability due to Liouville is inherited by invariant symplectic submanifolds is studied. In algebraic category for submanifolds of dimension $4$ such integrability is valid.
Citation: Sadetov S. T.,  On Algebraic Integrals of the Motion of Point over a Quadric in Quadratic Potential, Regular and Chaotic Dynamics, 2000, vol. 5, no. 2, pp. 201-212

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