The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere

DOI10.1134/S1560354713030064zbMath1367.70007OpenAlexW2143047168MaRDI QIDQ359005

Alexey V. Borisov, Ivan. A. Bizyaev, Ivan. S. Mamaev

Publication date: 9 August 2013

Published in: Regular and Chaotic Dynamics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1134/s1560354713030064

zbMATH Keywords

integrabilityinvariant measurereversibleinvolutionfirst integralnonholonomic constraintconformally Hamiltonian systemrubber rollingtensor invariant

Mathematics Subject Classification ID

Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests (37J35) Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics (70H33) Motion of a rigid body in contact with a solid surface (70E18) Nonholonomic dynamical systems (37J60)

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Cites Work

• Unnamed Item
• Unnamed Item
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• Unnamed Item
• How to control the Chaplygin ball using rotors. II
• Impulsive control of a symmetric ball rolling without sliding or spinning
• On the similarity of Hamiltonian and reversible vector fields in 4D
• Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems
• Chaplygin ball over a fixed sphere: an explicit integration
• Hamiltonization of the generalized Veselova LR system
• Rubber rolling over a sphere
• Routh symmetry in the Chaplygin's rolling ball
• On invariant manifolds of nonholonomic systems
• Two non-holonomic integrable problems tracing back to Chaplygin
• How to control Chaplygin's sphere using rotors
• Breakdown of symmetry in reversible systems
• Non-holonomic Chaplygin systems
• Poisson structure of the reversible \(1:1\) resonance
• Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
• Reversible systems
• On realizing nonholonomic constraints by viscous friction forces and Celtic stones stability
• Rolling of a rigid body on a plane and sphere. Hierarchy of dynamics
• Rolling of a ball on a surface. New integrals and hierarchy of dynamics
• New cases when the invariant measure and first integrals exist in the problem of a body rolling on a surface
• Rolling of a ball without spinning on a plane: the absence of an invariant measure in a system with a complete set of integrals
• Hamiltonization of non-holonomic systems in the neighborhood of invariant manifolds
• Rolling of a homogeneous ball over a dynamically asymmetric sphere
• The nonexistence of an invariant measure for an inhomogeneous ellipsoid rolling on a plane
• Rolling of a symmetric sphere on a horizontal plane without sliding or spinning
• Periodic flows, rank-two Poisson structures, and nonholonomic mechanics
• Rolling bodies with regular surface: controllability theory and applications
• Rubber Rolling: Geometry and Dynamics of 2-3-5 Distributions
• A Treatise on the Analytical Dynamics of Particles and Rigid Bodies
• Reversible Diffeomorphisms and Flows
• Poisson structures for reduced non-holonomic systems
• On the Routh sphere problem
• Rolling of a Rigid Body Without Slipping and Spinning: Kinematics and Dynamics

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