Description 
1 online resource (xvi, 404 pages) : illustrations (some color) 
Series 
Advanced series in nonlinear dynamics ; v. 26 

Advanced series in nonlinear dynamics ; v. 26

Contents 
Nonholonomically constrained motions. Newton's equations  Constraints  Lagranged'Alembert equations  Lagrange derivative  Hamiltond'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 
Summary 
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 trailblazers in their respective fields 
Bibliography 
Includes bibliographical references (pages 387393) and index 
Notes 
Print version record 
Subject 
Caratheodory measure.


Geometry, Differential.


Nonholonomic dynamical systems.


Rigidity (Geometry)

Form 
Electronic book

Author 
Duistermaat, J. J. (Johannes Jisse), 1942


Śniatycki, Jędrzej.

ISBN 
9789814289498 (electronic bk.) 

9814289493 (electronic bk.) 
