Chaplygin problem

Abstract: We consider a novel mechanical system consisting of two spherical bodies rolling over each other, which is a natural extension of the famous Chaplygin problem of rolling motion of a ball on a plane. In contrast to the previously explored non-holonomic systems, this one has a higher dimension and is considerably more complicated. One remarkable property of our system is the existence of “clandestine” linear in momenta first integrals. For a more trivial integrable system, their counterparts were discovered by Chaplygin. We have also found a few cases of integrability.

Abstract: This paper is concerned with the figures of equilibrium of a self-gravitating ideal fluid with a stratified density and a steady-state velocity field. As in the classical formulation of the problem, it is assumed that the figures, or their layers, uniformly rotate about an axis fixed in space. It is shown that the ellipsoid of revolution (spheroid) with confocal stratification, in which each layer rotates with a constant angular velocity, is at equilibrium. Expressions are obtained for the gravitational potential, change in the angular velocity and pressure, and the conclusion is drawn that the angular velocity on the outer surface is the same as that of the corresponding Maclaurin spheroid. We note that the solution found generalizes a previously known solution for piecewise constant density distribution. For comparison, we also present a solution, due to Chaplygin, for a homothetic density stratification. We conclude by considering a homogeneous spheroid in the space of constant positive curvature. We show that in this case the spheroid cannot rotate as a rigid body, since the angular velocity distribution of fluid particles depends on the distance to the symmetry axis.

Abstract: Quasi-Chaplygin Equations and Their Solutions Introduction The Linear Approximation Symmetric Self-Similar Solutions Ellipse-like and Ellipsoid-like Solutions Three Forms of the Quasi-Chaplygin Equations General Theory in the One-Dimensional Case System Reduction to the Laplace Equation The Evolutionary Selection Principle of Spontaneous Solutions The Four Simplest Perturbations Maxima and Minima of Density Profiles Parametric Representation of the Coordinate Length-Periodic Perturbations Hill-Shaped, Well-Shaped, and Wave-Shaped Solitary Perturbations Elementary Algebraic Solutions Concerning the Flow Integral and Particle Distribution Functions A Solution to the General Cauchy Problem The Media of the Chaplygin Gas Type The Chaplygin Gas The Bunemann Plasma Instability Tearing Instability of Plasma Current Sheet Parametric Plasma Instability in an External Field Heating-Radiational Instability of Plasma in an Electric Field Media of the "Drops on the Ceiling" Type Overturned Shallow Water Instability of a Cold Surface Stream in the Ocean Self-Focusing of Light in a Nonlinear Medium Modulation Instability of the Langmuir Waves in a Plasma Self-Contraction of Wave Packets and the Lighthill Criterion Instability of Waves in Deep Water and Instability of a Gravitating Gas Slab Separation of an Electron Beam into Bunches Separation of an Electron Beam into Slabs and Filaments A Model Mechanism for the Birth of Galaxies Media with Negative Azimuthal Numbers Decay of a Liquid Jet into Drops Plasma Pinch Gas-Dynamic Particle Acceleration in a Pinch Electrodynamic Particle Acceleration in a Pinch and Neutrons Relativistic Plasma Pinch as a Source of Cosmic Rays Long-Wave Perturbations of Solitons and Solitrons The Korteweg-de Vries (KdV) Solitons The Knoidal KdV Waves Two-Dimensional Solitons of the Kadomtsev-Petviashvili Equation Solitons of Benjamin-Ono and of Sine-Gordon Equations Instability of Solitrons of the Generalized KdV Equation Geometric Optics Approximation for Solitons Solitons of the Nonlinear Schrodinger Equation (NLSE) and Solitrons of the Generalized NLSE Energy Principle of Stability in the Theory of Generalized NLSE Solitrons Nonlinear Description of Hydrodynamic Discontinuity Instabilities The Tangential Velocity Discontinuity Instability Problems on TVDI with Bounding Walls The Corrugation Instability of Shock Waves Flame Front Instability An Equation for the Front of Solidification of a Fluid Stationary Patterns on the Front of Solidification A Mechanism for Development of Column-like Separateness and "Heaps of Logs" in Rocks Supplementary Topics in the Theory of Unstable Media The Chaplygin Problem and Its Solution Method Inclusion of the Dispersion Corrections to the Chaplygin Media A List of Equations with the Hilbert Operator Tables of Formulas and Functions Short List of Results Bibliography Index