Regular and Chaotic Dynamics最新文献

We prove the integrability of magnetic geodesic flows of (SO(n))-invariant Riemannian metrics on the rank two Stefel variety (V_{n,2}) with respect to the magnetic field (eta dalpha), where (alpha) is the standard contact form on (V_{n,2}) and (eta) is a real parameter.Also, we prove the integrability of magnetic sub-Riemannian geodesic flows for (SO(n))-invariant sub-Riemannian structures on (V_{n,2}). All statements in the limit (eta=0) imply the integrability of the problems without the influence of the magnetic field. We also consider integrable pendulum-type natural mechanical systems with the kinetic energy defined by (SO(n)times SO(2))-invariant Riemannian metrics. For (n=3), using the isomorphism (V_{3,2}cong SO(3)), the obtained integrable magnetic models reduce tointegrable cases of the motion of a heavy rigid body with a gyrostat around a fixed point:the Zhukovskiy – Volterra gyrostat, the Lagrange top with a gyrostat, and the Kowalevskitop with a gyrostat. As a by-product we obtain the Lax presentations for the Lagrangegyrostat and the Kowalevski gyrostat in the fixed reference frame (dual Lax representations).

This paper provides an original rendition of the heavy top that unravels the mysteries behind S. Kowalewski’s seminal work on the motions of a rigid body around a fixed point under the influence of gravity.The point of departure for understanding Kowalewski’s workbegins with Kirchhoff’s model for the equilibrium configurations of an elastic rod in ({mathbb{R}}^{3}) subject to fixed bending and twisting moments at its ends [17]. This initial orientation to the elastic problem shows, first, that the Kowalewski type integrals discovered by I. V. Komarov and V. B. Kuznetsov [24, 25] appear naturally on the Lie algebras associated with the orthonormal frame bundles of the sphere (S^{3}) and the hyperboloid (H^{3}) [17] and, secondly, it showsthat these integrals of motion can be naturally extracted from a canonical Poisson system on the dual of (so(4,mathbb{C})) generated byan affine quadratic Hamiltonian (H) (Kirchhoff – Kowalewski type).

The paper shows that the passage to complex variablesis synonymous with the representation of (so(4,mathbb{C})) as (sl(2,mathbb{C})times sl(2,mathbb{C})) and the embedding of (H) into (sp(4,mathbb{C})), an important intermediate step towards uncovering the origins of Kowalewski’s integral. There is a quintessential Kowalewski type integral of motion on (sp(4,mathbb{C})) that appears as a spectral invariant for the Poisson system associated with a Hamiltonian (mathcal{H}) (a natural extension of (H)) that satisfies Kowalewski’s conditions.

The text then demonstrates the relevance of this integral of motion for other studies in the existing literature [7, 35]. The text also includes a self-contained treatment of the integration of the Kowalewski type equations based on Kowalewski’s ingenuous separation of variables, the hyperelliptic curve and the solutions on its Jacobian variety.

The Korteweg – de Vries – Burgers equation, the modified Korteweg – de Vries – Burgers equation, the dissipative Gardner equation and the nonlinear differential equation for description surface waves in a convecting fluid are considered. The Cauchy problems for all these partial differential equations are not solved by theinverse scattering transform. Reductions of these equations to nonlinear ordinary differentialequations do not pass the Painlevé test. However, there are local expansions of the generalsolutions in the Laurent series near movable singular points.We demonstrate that the obtained information from the Painlevé test for reductions ofnonlinear evolution dissipative differential equations can be used to construct thenonautonomous first integrals of nonlinear ordinary differential equations. Taking intoaccount the found first integrals, we also obtain analytical solutions of nonlinear evolutiondissipative differential equations. Our approach is illustrated to obtain thenonautonomous first integrals for reduction of the Korteweg – de Vries – Burgers equation,the modified Korteweg – de Vries – Burgers equation, the dissipative Gardner equation andthe nonlinear differential equation for description surface waves in a convecting fluid.The obtained first integrals are used to construct exact solutions of the above-mentionednonlinear evolution equations with as many arbitrary constants as possible. It is shown thatsome exact solutions of the equation for description of nonlinear waves in a convectingliquid are expressed via the Painlevé transcendents.

The circular restricted three-body problem is used as an approximate model in space mission planning. Its periodic solutions around equilibrium points, which are referred to as the libration points, are utilized for exploration of possible spacecraft trajectories in the preliminary stages of mission design. In this paper, a numerical methodology for periodic libration point orbits (LPOs) computation is introduced and applied to the construction and study of N-periodic (up to (N=6)) quasi-planar orbit families in the Earth-Moon system. The stability and the bifurcation points of these families are determined. The proposed method is based on an iterative algorithm searching for the initial state vector of periodic LPOs, which allows computing unstable long-periodic and large-amplitude orbits. The method is suited to perform a straightforward switch to bifurcating branches of periodic orbits.

We consider motion of a material point placed in a constant homogeneous magnetic field in (mathbb{R}^{n}) and also motion restricted to the sphere (S^{n-1}).While there is an obvious integrability of the magnetic system in (mathbb{R}^{n}), the integrability of the system restricted to the sphere (S^{n-1}) is highly nontrivial. We provecomplete integrability of the obtained restricted magnetic systems for (nleqslant 6). The first integrals of motion of the magnetic flows on the spheres (S^{n-1}), for (n=5) and (n=6), are polynomials of degree(1), (2), and (3) in momenta.We prove noncommutative integrability of the obtained magnetic flows for any (ngeqslant 7) when the systems allow a reduction to the cases with (nleqslant 6). We conjecture that the restricted magnetic systems on (S^{n-1}) are integrable for all (n).

This paper addresses the spatial restricted elliptic problemof three bodies (material points) gravitating toward each other under Newton’s law ofgravitation. The eccentricity of the orbit of the main attracting bodies is assumed to besmall, and nonlinear oscillations ofa passively gravitating body near a Lagrangian triangular libration point are studied.It is assumed that in the limiting case of the circular problem the ratioof the frequency of rotation of the main bodies about their common center of massto the value of one of the frequencies of small linear oscillations of the passive bodyis exactly equal to three. A detailed analysis is made of two different particular cases ofinfluence of the three-dimensionality of theproblem on the characteristics of nonlinear oscillations of the passive body.

We show that the equations of motion of a rigid body about a fixed point in the Newtonianfield with a quadratic potential are special reduction of period-one closure of the Darboux dressing chain for the Schrödinger operators with matrix potentials. Some new explicit solutions of the corresponding matrix system and the spectral properties of the related Schrödinger operators are discussed.

We explicitly compute the maximal Lyapunov exponent for a switched system on (mathrm{SL}_{2}(mathbb{R})) and the corresponding switching function which realizes the maximal exponent. This computation is reduced to the characterization of optimal trajectories for an optimal control problem on the Lie group.

The aim of this note is to present a simple observation that a slight refinement of thecontraction mapping principle allows one to recover the precise convergence rate in thePicard – Lindelöf theorem.