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Vol 24, No 4 (2019)
- Year: 2019
- Articles: 5
- URL: https://ogarev-online.ru/1560-3547/issue/view/13604
Article
The Kepler Problem: Polynomial Algebra of Nonpolynomial First Integrals
Abstract
The sum of elliptic integrals simultaneously determines orbits in the Kepler problem and the addition of divisors on elliptic curves. Periodic motion of a body in physical space is defined by symmetries, whereas periodic motion of divisors is defined by a fixed point on the curve. The algebra of the first integrals associated with symmetries is a well-known mathematical object, whereas the algebra of the first integrals associated with the coordinates of fixed points is unknown. In this paper, we discuss polynomial algebras of nonpolynomial first integrals of superintegrable systems associated with elliptic curves.
353-369
Singular Reduction of the 2-Body Problem on the 3-Sphere and the 4-Dimensional Spinning Top
Abstract
We consider the dynamics and symplectic reduction of the 2-body problem on a sphere of arbitrary dimension. It suffices to consider the case when the sphere is 3-dimensional. As the 3-sphere is a group it acts on itself by left and right multiplication and these together generate the action of the SO(4) symmetry on the sphere. This gives rise to a notion of left and right momenta for the problem, and allows for a reduction in stages, first by the left and then the right, or vice versa. The intermediate reduced spaces obtained by left or right reduction are shown to be coadjoint orbits of the special Euclidean group SE(4). The full reduced spaces are generically 4-dimensional and we describe these spaces and their singular strata. The dynamics of the 2-body problem descend through a double cover to give a dynamical system on SO(4) which, after reduction and for a particular choice of Hamiltonian, coincides with that of a 4-dimensional spinning top with symmetry. This connection allows us to “hit two birds with one stone” and derive results about both the spinning top and the 2-body problem simultaneously. We provide the equations of motion on the reduced spaces and fully classify the relative equilibria and discuss their stability.
370-391
On Transversal Connecting Orbits of Lagrangian Systems in a Nonstationary Force Field: the Newton – Kantorovich Approach
Abstract
We consider a natural Lagrangian system defined on a complete Riemannian manifold subjected to the action of a nonstationary force field with potential U(q,t) = f(t)V(q). It is assumed that the factor f(t) tends to ∞ as t → ±∞ and vanishes at a unique point t0 ∈ ℝ. Let X+, X− denote the sets of isolated critical points of V(x) at which U(x,t) as a function of x attains its maximum for any fixed t > t0 and t < t0, respectively. Under nondegeneracy conditions on points of X± we apply the Newton – Kantorovich type method to study the existence of transversal doubly asymptotic trajectories connecting X− and X+. Conditions on the Riemannian manifold and the potential which guarantee the existence of such orbits are presented. Such connecting trajectories are obtained by continuation of geodesies defined in a vicinity of the point t0 to the whole real line.
392-417
Bifurcation Diagram of One Generalized Integrable Model of Vortex Dynamics
Abstract
This article is devoted to the results of phase topology research on a generalized mathematical model, which covers such two problems as the dynamics of two point vortices enclosed in a harmonic trap in a Bose – Einstein condensate and the dynamics of two point vortices bounded by a circular region in an ideal fluid. New bifurcation diagrams are obtained and three-into-one and four-into-one tori bifurcations are observed for some values of the physical parameters of the model. The presence of such bifurcations in the integrable model of vortex dynamics with positive intensities indicates a complex transition and a connection between bifurcation diagrams in both limiting cases. In this paper, we analytically derive equations that define the parametric family of bifurcation diagrams of the generalized model, including bifurcation diagrams of the specified limiting cases. The dynamics of the bifurcation diagram in a general case is shown using its implicit parameterization. A stable bifurcation diagram, related to the problem of dynamics of two vortices bounded by a circular region in an ideal fluid, is observed for particular parameters’ values.
418-431
Impact of Noise on the Amplitude Chimera Lifetime in an Ensemble of Nonlocally Coupled Chaotic Maps
Abstract
This paper presents results of numerical statistical analysis of the effect of short-term localized noise of different intensity on the amplitude chimera lifetime in an ensemble of nonlocally coupled logistic maps in a chaotic regime. It is shown that a single and rather weak noise perturbation added only to the incoherence cluster of the amplitude chimera after its switching to the phase chimera mode is able to revive and stabilize the amplitude chimera, as well as to increase its lifetime to infinity. It is also analyzed how the amplitude chimera lifetime depends on the duration of noise influence of different intensity.
432-445