Geometry of nonholonomically constrained systems / Richard Cushman, Hans Duistermaat, Jędrzej Śniatycki.Material type: TextSeries: Advanced series in nonlinear dynamics ; v. 26.Publication details: New Jersey : World Scientific, ©2010. Description: 1 online resource (xvi, 404 pages) : illustrations (some color)Content type: text Media type: computer Carrier type: online resourceISBN: 9789814289498; 9814289493Subject(s): Geometry, Differential | Nonholonomic dynamical systems | Rigidity (Geometry) | Caratheodory measure | MATHEMATICS -- Geometry -- Differential | Caratheodory measure | Geometry, Differential | Nonholonomic dynamical systems | Rigidity (Geometry) | Differentialgeometrie -- Mechanik | Mechanik -- Differentialgeometrie | Geometry | Mathematics | Physical Sciences & MathematicsGenre/Form: Electronic book. Additional physical formats: Print version:: Geometry of nonholonomically constrained systems.DDC classification: 516.3/6 LOC classification: QA614.833 | .C87 2010ebOnline resources: Click here to access online
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This book gives a modern differential geometric treatment of linearly nonholonomically constrained systems. It discusses in detail what is meant by symmetry of such a system and gives a general theory of how to reduce such a symmetry using the concept of a differential space and the almost Poisson bracket structure of its algebra of smooth functions. The above theory is applied to the concrete example of Carathéodory's sleigh and the convex rolling rigid body. The qualitative behavior of the motion of the rolling disk is treated exhaustively and in detail. In particular, it classifies all motions of the disk, including those where the disk falls flat and those where it nearly falls flat. The geometric techniques described in this book for symmetry reduction have not appeared in any book before. Nor has the detailed description of the motion of the rolling disk. In this respect, the authors are trail-blazers in their respective fields.
Includes bibliographical references (pages 387-393) and index.
Nonholonomically constrained motions. Newton's equations -- Constraints -- Lagrange-d'Alembert equations -- Lagrange derivative -- Hamilton-d'Alembert equations -- Distributional Hamiltonian formulation -- Almost Poisson brackets -- Momenta and momentum equation -- Projection principle -- Accessible sets -- Constants of motion -- Notes -- Group actions and orbit spaces. Group actions -- Orbit spaces -- Isotropy and orbit types -- Smooth structure on an orbit space -- Subcartesian spaces -- Stratification of the orbit space by orbit types -- Derivations and vector fields on a differential space -- Vector fields on a stratified differential space -- Vector fields on an orbit space -- Tangent objects to an orbit space -- Notes -- Symmetry and reduction. Dynamical systems with symmetry -- Nonholonomic singular reduction -- Nonholonomic regular reduction -- Chaplygin systems -- Orbit types and reduction -- Conservation laws -- Lifted actions and the momentum equation -- Notes -- Reconstruction, relative equilibria and relative periodic orbits. Reconstruction -- Relative equilibria -- Relative periodic orbits -- Notes -- Carathéodory's sleigh. Basic set up -- Equations of motion -- Reduction of the E(2) symmetry -- Motion on the E(2) reduced phase space -- Reconstruction -- Notes -- Convex rolling rigid body. Basic set up -- Unconstrained motion -- Constraint distribution -- Constrained equations of motion -- Reduction of the translational [symbol] symmetry -- Reduction of E(2) symmetry -- Body of revolution -- Notes -- The rolling disk. General set up -- Reduction of the E(2) x S[symbol] symmetry -- Reconstruction -- Relative equilibria -- A potential function on an interval -- Scaling -- Solutions of the rescaled Chaplygin equations -- Bifurcations of a vertical disk -- The global geometry of the degeneracy locus -- Falling flat. -- Near falling flat -- The bifurcation diagram -- The integral map -- Constant energy slices -- The spatial rotational shift -- Notes.
Print version record.