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ON A MAPPING PROBLEM BETWEEN 3D EUCLIDEAN AND 3DRIEMANNIAN SPACES

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As is well known, the temporal evolution of a classical system is uniquely determined by the Hamilton equations, which form a system of ordinary second-order differential equations. The mathematical problem is to integrate this system of differential equations and find all possible functions of variable ”t” (usual time), which, after substituting into the equations, turns them into an identity. In the case of dynamical Poincar´e systems, the system of equations, as a rule, cannot be fully integrated, since the number of integrals of motion is often less than the number of degrees of freedom. In a series of papers [1, 2, 3], using the classical three-body problem as an example, it was shown that at formulation the problem on Riemannian geometry new hidden symmetries of the dynamical system are revealed. This allows us to make the integration of the problem more complete. Note that in the formulation of the dynamical problem on a Riemannian manifold, the key role is played by the proof of the homeomorphism theorem between the 3D Euclidean subspace E 3 and the 3D conformal Euclidean space M(3) . However, in the proof of the theorem an important role is played by the auxiliary manifold (see Fig. 1), which arises as a result of the mapping subspace E 3 onto the manifold M(3) . In particular, it was shown that the corresponding system of underdetermined

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Institute for Informatics and Automation Problems of NAS RA ; Institute of Chemical Physics, NAS of RA

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ՄԻՋԱԶԳԱՅԻՆ ԳԻՏԱԺՈՂՈՎՆՎԻՐՎԱԾ ԵՐԵՎԱՆԻ ՊԵՏԱԿԱՆ ՀԱՄԱԼՍԱՐԱՆԻ100-ԱՄՅԱԿԻՆ