Contents -- Acknowledgments -- Foreword -- 1. Nonholonomically constrained motions -- 1.1 Newton's equations -- 1.2 Constraints -- 1.3 Lagrange-d'Alembert equations -- 1.4 Lagrange derivative in a trivialization -- 1.5 Hamilton-d'Alembert equations -- 1.6 Distributional Hamiltonian formulation -- 1.6.1 The symplectic distribution (H,) -- 1.6.2 H and in a trivialization -- 1.6.3 Distributional Hamiltonian vector field -- 1.7 Almost Poisson brackets -- 1.7.1 Hamilton's equations -- 1.7.2 Nonholonomic Dirac brackets -- 1.8 Momenta and momentum equation -- 1.8.1 Momentum functions -- 1.8.2 Momentum equations -- 1.8.3 Homogeneous functions -- 1.8.4 Momenta as coordinates -- 1.9 Projection principle -- 1.10 Accessible sets -- 1.11 Constants of motion -- 1.12 Notes -- 2. Group actions and orbit spaces -- 2.1 Group actions -- 2.2 Orbit spaces -- 2.3 Isotropy and orbit types -- 2.3.1 Isotropy types -- 2.3.2 Orbit types -- 2.3.3 When the action is proper -- 2.3.4 Stratification on by orbit types -- 2.4 Smooth structure on an orbit space -- 2.4.1 Differential structure -- 2.4.2 The orbit space as a differential space -- 2.5 Subcartesian spaces -- 2.6 Stratification of the orbit space by orbit types -- 2.6.1 Orbit types in an orbit space -- 2.6.2 Stratification of an orbit space -- 2.6.3 Minimality of S -- 2.7 Derivations and vector fields on a differential space -- 2.8 Vector fields on a stratified differential space -- 2.9 Vector fields on an orbit space -- 2.10 Tangent objects to an orbit space -- 2.10.1 Stratified tangent bundle -- 2.10.2 Zariski tangent bundle -- 2.10.3 Tangent cone -- 2.10.4 Tangent wedge -- 2.11 Notes -- 3. Symmetry and reductio -- 3.1 Dynamical systems with symmetry -- 3.1.1 Invariant vector fields -- 3.1.2 Reduction of symmetry -- 3.1.3 Reduction for or a free and proper G-action -- 3.1.4 Reduction of a nonfree, proper G-action.

3.2 Nonholonomic singular reduction for a proper action -- 3.3 Nonholonomic reduction for a free and proper action -- 3.4 Chaplygin systems -- 3.5 Orbit types and reduction -- 3.6 Conservation laws -- 3.6.1 Momentum map -- 3.6.2 Gauge momenta -- 3.7 Lifted actions and the momentum equation -- 3.7.1 Lifted actions -- 3.7.2 Momentum equation -- 3.8 Notes -- 4.Reconstruction, relative equilibria and relative periodic orbits -- 4.1 Reconstruction -- 4.1.1 Reconstruction for proper free actions -- 4.1.2 Reconstruction for nonfree proper actions -- 4.1.3 Application to nonholonomic systems -- 4.2 Relative equilibria -- 4.2.1 Basic properties -- 4.2.2 Quasiperiodic relative equilibria -- 4.2.3 Runaway relative equilibria -- 4.2.4 Relative equilibria when the action is not free -- 4.2.5 Other relative equilibria in a G-orbit -- 4.2.5.1 When the G-action is free -- 4.2.5.2 When the G-action is not free -- 4.2.6 Smooth families of quasiperiodic relative equilibria -- 4.2.6.1 Elliptic, regular, and stably elliptic elements of g -- 4.2.6.2 When the G-action is free and proper -- 4.2.6.3 When the G-action is proper but not free -- 4.3 Relative periodic orbits -- 4.3.1 Basic properties -- 4.3.2 Quasiperiodic relative periodic orbits -- 4.3.3 Runaway relative period orbits -- 4.3.4 When the G-action is not free -- 4.3.5 Other relative periodic orbits in the (G × R)-orbit -- 4.3.6 Smooth families of quasiperiodic relative periodic orbits -- 4.3.6.1 Elliptic, regular, and stably elliptic elements of G -- 4.3.6.2 When the G-action is free -- 4.3.6.3 When the G-action is not free -- 4.4 Notes -- 5. Caratheodory's sleigh -- 5.1 Basic set up -- 5.1.1 Configuration space -- 5.1.2 Kinetic energy -- 5.1.3 Nonholonomic constraint -- 5.2 Equations of motion -- 5.2.1 Lagrange-d'Alembert equations -- 5.2.2 Nonholonomic Dirac brackets.

5.2.3 Lagrange-d' Alembert in a trivialzation -- 5.2.4 Almost Poisson bracket form -- 5.2.4.1 H and in a trivialization -- 5.2.4.2 Almost Poisson bracket of c1 and c2 -- 5.2.4.3 Equations of motion -- 5.2.5 Distributional Hamiltonian system -- 5.2.5.1 Lie group model -- 5.2.5.2 The distribution H and its symplectic form H -- 5.2.5.3 Equations of motion -- 5.3 Reduction of the E(2) symmetry -- 5.3.1 The E(2) symmetry -- 5.3.2 The momentum equation -- 5.3.3 E(2)-reduced equations of motion -- 5.3.3.1 The E(2)-reduced equations using almost Poisson brackets -- 5.4 Motion the E(2) reduced phase space -- 5.5 Reconstruction -- 5.5.1 Relative equilibria -- 5.5.2 General motions -- 5.5.3 Motion of a material point on the sleigh -- 5.6 Notes -- 6. Convex rolling rigid body -- 6.1 Basic set up -- 6.2 Unconstrained motion -- 6.3 Constraint distribution -- 6.4 Constrained equations of motion -- 6.4.1 Vector field on D -- 6.4.2 Computation of H and in a trivialization -- 6.4.3 Distributional vector field in a trivialization -- 6.5 Reduction of the translational R2 symmetry -- 6.5.1 The R2-reduced equations of motion -- 6.5.2 Comparison with the Euler-Lagrange equations -- 6.5.3 The R2-reduced distribution HDN and the 2-form -- 6.6 Reduction of E(2) symmetry -- 6.6.1 E(2) symmetry -- 6.6.2 E(2) (2)-orbit space -- 6.6.3 E(2)-reduced distribution and 2-form -- 6.6.4 Reduced distributional system -- 6.7 Body of revolution -- 6.7.1 Geometric and dynamic symmetry -- 6.7.2 Reduction of the induced axial symmetry -- 6.7.3 Axially reduced equations of motion -- 6.7.3.1 Chaplygin equation -- 6.7.3.2 Two additional constants of motion -- 6.7.3.3 The total energy -- 6.7.3.4 A conservatative Newtonian system -- 6.7.3.5 Solution of the one degree of freedom system -- 6.7.3.6 Quasi-periodic solutions -- 6.7.3.7 Appendix. The E(2) × S1-reduced equations of motion.

6.8 Notes -- 7. The rolling disk -- Summary -- 7.1 General set up -- 7.2 Reduction of the E(2) × S1 symmetry -- 7.2.1 First E(2), then S1 -- 7.2.2 First S1, then E(2) -- 7.3 Reconstruction -- 7.3.1 The E(2)-reduced flow -- 7.3.2 The full motion -- 7.3.3 The S1-reduced flow -- 7.3.4 Geometry of the E(2) × S1 reduction map -- 7.4 Relative equilibria -- 7.4.1 The manifold of relative equilibria -- 7.4.2 One parameter groups -- 7.4.3 Angular speeds in terms of invariants -- 7.4.4 Motion of the relative equilibria -- 7.4.5 Nearly flat relative equilibria -- 7.5 A potential function on an interval -- 7.5.1 Chaplygin's equations -- 7.5.2 A conservative Newtonian system -- 7.5.3 Qualitative behavior -- 7.5.4 A special case of falling flat -- 7.6 Scaling -- 7.7 Solutions of the rescaled Chaplygin equations -- 7.7.1 The recessive solution -- 7.7.2 Asymptotics -- 7.7.3 The normalized even and odd solutions -- 7.7.4 Computation of r(0) and r0(0) -- 7.8 Bifurcations of a vertical disk -- 7.8.1 Degenerate equilibria -- 7.8.2 Vertical degenerate relative equilibria -- 7.8.3 Normal form of the potential -- 7.8.4 Cusps of the degeneracy locus -- 7.9 The global geometry of the degeneracy locus -- 7.9.1 The circle of degenerate critical points -- 7.9.2 A global description of the degeneracy locus -- 7.10 Falling flat -- 7.10.1 When the disk does not fall flat -- 7.10.2 When the disk falls flat -- 7.10.3 Limiting behavior when falling flat -- 7.11 Near falling flat -- 7.11.1 Elastic reflection -- 7.11.2 The increase of the angles and -- 7.11.3 Motions near falling flat -- 7.12 The bifurcation diagram -- 7.12.1 The bifurcation set B -- 7.12.2 Off the bifurcation set B -- 7.12.3 On a coordinate axis or in an open quadrant -- 7.12.4 Near `± -- 7.12.5 Global qualitative description of V -- 7.12.6 Global description of the orbits of X -- 7.13 The integral map.

7.13.1 Regular values of I -- 7.13.2 The global geometry of the critical value surface -- 7.13.2.1 The critical value surface -- 7.13.2.2 The singularities of -- 7.13.2.3 The global structure of -- 7.14 Constant energy slices -- 7.14.1 Numerical pictures of the constant energy slices -- 7.14.2 Geometric features of the constant energy slices -- 7.14.3 Outward radial growth -- 7.14.4 The swallow tail sections -- 7.14.5 Behavior of cusp points -- 7.14.6 Over the coordinate axes in the ( 3, 4)-plane -- 7.14.7 over `± -- 7.15 The spatial rotational shift -- 7.15.1 The shift -- 7.15.2 Quasiperiodic motion -- 7.15.3 The spatial rotational shift -- 7.15.4 Near elliptic relative equilibria -- 7.15.5 Nearly flat solutions -- 7.16 Notes -- Bibliography -- Index.