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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 mot

Nonholonomic dynamical systems. --- Geometry, Differential. --- Rigidity (Geometry) --- Caratheodory measure. --- Measure, Caratheodory --- Algebra, Boolean --- Measure theory --- Geometric rigidity --- Rigidity theorem --- Discrete geometry --- Differential geometry --- Dynamical systems, Nonholonomic --- Non-holonomic systems --- Nonholonomic systems --- Differentiable dynamical systems --- Rigidity (Geometry).

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"This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners), analogous to the toric case, but their associated integral affine structures are singular, with non-trivial monodromy, due to focus singularities. We obtain a series of convexity results, both positive and negative, for such singular integral affine base spaces. In particular, near a focus singular point, they are locally convex and the local-global convexity principle still applies. They are also globally convex under some natural additional conditions. However, when the monodromy is sufficiently large, the local-global convexity principle breaks down and the base spaces can be globally non-convex, even for compact manifolds. As a surprising example, we construct a 2-dimensional "integral affine black hole", which is locally convex but for which a straight ray from the center can never escape"--

Convex domains. --- Affine differential geometry. --- Hamiltonian systems. --- Toric varieties. --- Dynamical systems and ergodic theory -- Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems -- Completely integrable systems, topological structure of phase space, integratio --- Differential geometry -- Classical differential geometry -- Affine differential geometry. --- Convex and discrete geometry -- General convexity -- Axiomatic and generalized convexity.

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Ergodic theory. Information theory --- Ordinary differential equations --- 514.8 --- Geometric study of objects of mechanics and physics --- 514.8 Geometric study of objects of mechanics and physics --- Nonholonomic dynamical systems. --- Hamiltonian systems. --- Lagrangian points. --- Hamiltonian systems --- Lagrangian points --- Nonholonomic dynamical systems --- Dynamical systems, Nonholonomic --- Non-holonomic systems --- Nonholonomic systems --- Differentiable dynamical systems --- Equilibrium points, Lagrangian --- Lagrangian equilibrium points --- Libration points --- Points, Lagrangian --- Orbits --- Hamiltonian dynamical systems --- Systems, Hamiltonian

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Nonholonomic systems are control systems which depend linearly on the control. Their underlying geometry is the sub-Riemannian geometry, which plays for these systems the same role as Euclidean geometry does for linear systems. In particular the usual notions of approximations at the first order, that are essential for control purposes, have to be defined in terms of this geometry. The aim of these notes is to present these notions of approximation and their application to the motion planning problem for nonholonomic systems.

Nonholonomic dynamical systems. --- Geometry, Differential. --- Differential geometry --- Dynamical systems, Nonholonomic --- Non-holonomic systems --- Nonholonomic systems --- Differentiable dynamical systems --- Systems theory. --- Global differential geometry. --- Artificial intelligence. --- Mathematics. --- Computer science. --- Systems Theory, Control. --- Differential Geometry. --- Artificial Intelligence. --- Mathematics, general. --- Computer Science, general. --- Informatics --- Science --- Math --- AI (Artificial intelligence) --- Artificial thinking --- Electronic brains --- Intellectronics --- Intelligence, Artificial --- Intelligent machines --- Machine intelligence --- Thinking, Artificial --- Bionics --- Cognitive science --- Digital computer simulation --- Electronic data processing --- Logic machines --- Machine theory --- Self-organizing systems --- Simulation methods --- Fifth generation computers --- Neural computers --- Geometry, Differential --- System theory. --- Differential geometry. --- Systems, Theory of --- Systems science --- Philosophy

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A general approach to the derivation of equations of motion of as holonomic, as nonholonomic systems with the constraints of any order is suggested. The system of equations of motion in the generalized coordinates is regarded as a one vector relation, represented in a space tangential to a manifold of all possible positions of system at given instant. The tangential space is partitioned by the equations of constraints into two orthogonal subspaces. In one of them for the constraints up to the second order, the motion low is given by the equations of constraints and in the other one for ideal constraints, it is described by the vector equation without reactions of connections. In the whole space the motion low involves Lagrangian multipliers. It is shown that for the holonomic and nonholonomic constraints up to the second order, these multipliers can be found as the function of time, positions of system, and its velocities. The application of Lagrangian multipliers for holonomic systems permits us to construct a new method for determining the eigenfrequencies and eigenforms of oscillations of elastic systems and also to suggest a special form of equations for describing the system of motion of rigid bodies. The nonholonomic constraints, the order of which is greater than two, are regarded as programming constraints such that their validity is provided due to the existence of generalized control forces, which are determined as the functions of time. The closed system of differential equations, which makes it possible to find as these control forces, as the generalized Lagrange coordinates, is compound. The theory suggested is illustrated by the examples of a spacecraft motion. The book is primarily addressed to specialists in analytic mechanics.

Mathematics -- Differential Equations -- General. --- Mathematics. --- Nonholonomic dynamical systems. --- Nonholonomic dynamical systems --- Civil Engineering --- Geometry --- Civil & Environmental Engineering --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Differentiable dynamical systems. --- Dynamical systems, Nonholonomic --- Non-holonomic systems --- Nonholonomic systems --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Engineering. --- Mechanics. --- Computational intelligence. --- Mechanics, Applied. --- Theoretical and Applied Mechanics. --- Computational Intelligence. --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Intelligence, Computational --- Artificial intelligence --- Soft computing --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Construction --- Industrial arts --- Technology --- Differentiable dynamical systems --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Mechanics, applied. --- Classical Mechanics.