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Volume 27, Number 4

Volume 27, Number 4, 2022
Alexey Borisov Memorial Volume

Celletti A.,  Karampotsiou E.,  Lhotka C.,  Pucacco G.,  Volpi M.
The Role of Tidal Forces in the Long-term Evolution of the Galilean System
Abstract
The Galilean satellites of Jupiter are called Io, Europa, Ganymede and Callisto. The first three moons are found in the so-called Laplace resonance, which means that their orbits are locked in a 2 : 1 resonant chain. Dissipative tidal effects play a fundamental role, especially when considered on long timescales. The main objective of this work is the study of the persistence of the resonance along the evolution of the system when considering the tidal interaction between Jupiter and Io. To constrain the computational cost of the task, we enhance this dissipative effect by means of a multiplying factor. We develop a simplified model to study the propagation of the tidal effects from Io to the other moons, resulting in the outward migration of the satellites. We provide an analytical description of the phenomenon, as well as the behaviour of the semi-major axis of Io as a function of the figure of merit. We also consider the interaction of the inner trio with Callisto, using a more elaborated Hamiltonian model allowing us to study the long-term evolution of the system along few gigayears. We conclude by studying the possibility of the trapping into resonance of Callisto depending on its initial conditions.
Keywords: Laplace resonance, tidal dissipation, libration, normal form
Citation: Celletti A.,  Karampotsiou E.,  Lhotka C.,  Pucacco G.,  Volpi M., The Role of Tidal Forces in the Long-term Evolution of the Galilean System, Regular and Chaotic Dynamics, 2022, vol. 27, no. 4, pp. 381-408
DOI:10.1134/S1560354722040013
Fasso F.,  Sansonetto N.
On Some Aspects of the Dynamics of a Ball in a Rotating Surface of Revolution and of the Kasamawashi Art
Abstract
We study some aspects of the dynamics of the nonholonomic system formed by a heavy homogeneous ball constrained to roll without sliding on a steadily rotating surface of revolution. First, in the case in which the figure axis of the surface is vertical (and hence the system is $\textrm{SO(3)}\times\textrm{SO(2)}$-symmetric) and the surface has a (nondegenerate) maximum at its vertex, we show the existence of motions asymptotic to the vertex and rule out the possibility of blowup. This is done by passing to the 5-dimensional $\textrm{SO(3)}$-reduced system. The $\textrm{SO(3)}$-symmetry persists when the figure axis of the surface is inclined with respect to the vertical — and the system can be viewed as a simple model for the Japanese kasamawashi (turning umbrella) performance art — and in that case we study the (stability of the) equilibria of the 5-dimensional reduced system.
Keywords: nonholonomic mechanical systems with symmetry, rolling rigid bodies, relative equilibria, kasamawashi
Citation: Fasso F.,  Sansonetto N., On Some Aspects of the Dynamics of a Ball in a Rotating Surface of Revolution and of the Kasamawashi Art, Regular and Chaotic Dynamics, 2022, vol. 27, no. 4, pp. 409-423
DOI:10.1134/S1560354722040025
Dragović V.,  Gajić B.,  Jovanović B.
Spherical and Planar Ball Bearings — Nonholonomic Systems with Invariant Measures
Abstract
We first construct nonholonomic systems of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with centers $O_1,\ldots,O_n$ and with the same radius $r$ that are rolling without slipping around a fixed sphere $\mathbf S_0$ with center $O$ and radius $R$. In addition, it is assumed that a dynamically nonsymmetric sphere $\mathbf S$ of radius $R+2r$ and the center that coincides with the center $O$ of the fixed sphere $\mathbf S_0$ rolls without slipping over the moving balls $\mathbf B_1,\dots,\mathbf B_n$. We prove that these systems possess an invariant measure. As the second task, we consider the limit, when the radius $R$ tends to infinity. We obtain a corresponding planar problem consisting of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with centers $O_1,\ldots,O_n$ and the same radius $r$ that are rolling without slipping over a fixed plane $\Sigma_0$, and a moving plane $\Sigma$ that moves without slipping over the homogeneous balls. We prove that this system possesses an invariant measure and that it is integrable in quadratures according to the Euler – Jacobi theorem.
Keywords: nonholonimic dynamics, rolling without slipping, invariant measure, integrability
Citation: Dragović V.,  Gajić B.,  Jovanović B., Spherical and Planar Ball Bearings — Nonholonomic Systems with Invariant Measures, Regular and Chaotic Dynamics, 2022, vol. 27, no. 4, pp. 424-442
DOI:10.1134/S1560354722040037
Dawson S. R.,  Dullin H. R.,  Nguyen D. M.
The Harmonic Lagrange Top and the Confluent Heun Equation
Abstract
The harmonic Lagrange top is the Lagrange top plus a quadratic (harmonic) potential term. We describe the top in the space fixed frame using a global description with a Poisson structure on $T^*S^3$. This global description naturally leads to a rational parametrisation of the set of critical values of the energy-momentum map. We show that there are 4 different topological types for generic parameter values. The quantum mechanics of the harmonic Lagrange top is described by the most general confluent Heun equation (also known as the generalised spheroidal wave equation). We derive formulas for an infinite pentadiagonal symmetric matrix representing the Hamiltonian from which the spectrum is computed.
Keywords: symmetric rigid body, Lagrange top, Hamiltonian Hopf bifurcation, quantisation, confluent Heun equation
Citation: Dawson S. R.,  Dullin H. R.,  Nguyen D. M., The Harmonic Lagrange Top and the Confluent Heun Equation, Regular and Chaotic Dynamics, 2022, vol. 27, no. 4, pp. 443-459
DOI:10.1134/S1560354722040049
Kordyukov Y. A.,  Taimanov I. A.
Trace Formula for the Magnetic Laplacian on a Compact Hyperbolic Surface
Abstract
We compute the trace formula for the magnetic Laplacian on a compact hyperbolic surface of constant curvature with a constant magnetic field for energies above the Mane critical level of the corresponding magnetic geodesic flow. We discuss the asymptotic behavior of the coefficients of the trace formula when the energy approaches the Mane critical level.
Keywords: trace formula, magnetic Laplacian, magnetic geodesic flow
Citation: Kordyukov Y. A.,  Taimanov I. A., Trace Formula for the Magnetic Laplacian on a Compact Hyperbolic Surface, Regular and Chaotic Dynamics, 2022, vol. 27, no. 4, pp. 460-476
DOI:10.1134/S1560354722040050
Mendoza V.
The Dynamical Core of a Homoclinic Orbit
Abstract
The complexity of a dynamical system exhibiting a homoclinic orbit is given by its dynamical core which, due to Cantwell, Conlon and Fenley, is a set uniquely determined in the isotopy class, up to a topological conjugacy, of the end-periodic map relative to that orbit. In this work we prove that a sufficient condition to determine the dynamical core of a homoclinic orbit of a Smale diffeomorphism on the 2-disk is the non-existence of bigons relative to this orbit. Moreover, we propose a pruning method for eliminating bigons that can be used to find a Smale map without bigons and hence for finding the dynamical core.
Keywords: Homoclinic orbits, dynamical core, Smale horseshoe, pruning theory
Citation: Mendoza V., The Dynamical Core of a Homoclinic Orbit, Regular and Chaotic Dynamics, 2022, vol. 27, no. 4, pp. 477-491
DOI:10.1134/S1560354722040062

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