Dynamical systems VII : integrable systems, nonholonomic dynamical systems / V. I. Arnol'd, S. P. Novikov (eds.).
Idioma: Inglés Lenguaje original: Ruso Series Encyclopaedia of mathematical sciences ; v. 16Editor: Berlin : Springer-Verlag, c1994Descripción: 341 p. : il. ; 24 cmISBN: 0387181768 (New York); 3540181768 (Berlin)Otro título: Integrable systems, nonholonomic dynamical systemsTema(s): Differentiable dynamical systems | Nonholonomic dynamical systemsOtra clasificación: 58Fxx (58-06)Chapter 1. Geometry of Distributions [10] §1. Distributions and Related Objects [10] 1.1. Distributions and Differential Systems [10] 1.2. Frobenius Theorem and the Flag of a Distribution [12] 1.3. Codistributions and Pfaffian Systems [14] 1.4. Regular Distributions [16] 1.5. Distributions Invariant with Respect to Group Actions and Some Canonical Examples [18] 1.6. Connections as Distributions [21] 1.7. A Classification of Left Invariant Contact Structures on Three-Dimensional Lie Groups [22] §2. Generic Distributions and Sets of Vector Fields, and Degeneracies of Small Codimension. Nilpotentization and Classification Problem [24] 2.1. Generic Distributions [25] 2.2. Normal Forms of Jets of Basic Vector Fields of a Generic Distribution [26] 2.3. Degeneracies of Small Codimension [27] 2.4. Generic Sets of Vector Fields [29] 2.5. Small Codimension Degeneracies of Sets of Vector Fields [31] 2.6. Projection Map Associated with a Distribution [32] 2.7. Classificaton of Regular Distributions [33] 2.8. Nilpotentization and Nilpotent Calculus [34] Chapter 2. Basic Theory of Nonholonomic Riemannian Manifolds [35] § 1. General Nonholonomic Variational Problem and the Geodesic Flow on Nonholonomic Riemannian Manifolds [35] 1.1. Rashevsky-Chow Theorem and Nonholonomic Riemannian Metrics (Camot-Caratheodory Metrics) [35] 1.2. Two-Point Problem and the Hopf-Rinow Theorem [36] 1.3. The Cauchy Problem and the Nonholonomic Geodesic Flow [37] 1.4. The Euler-Lagrange Equations in Invariant Form and in the Orthogonal Moving Frame and Nonholonomic Geodesics [38] 1.5. The Standard Form of Equations of Nonholonomic Geodesics for Generic Distributions [40] 1.6. Nonholonomic Exponential Mapping and the Wave Front [41] 1.7. The Action Functional [42] §2. Estimates of the Accessibility Set [42] 2.1. The Parallelotope Theorem [43] 2.2. Polysystems and Finslerian Metrics [44] 2.3. Theorem on the Leading Term [45] 2.4. Estimates of Generic Nonholonomic Metrics on Compact Manifolds [47] 2.5. Hausdorff Dimension of Nonholonomic Riemannian Manifolds [48] 2.6. The Nonholonomic Ball in the Heisenberg Group as the Limit of Powers of a Riemannian Ball [49] Chapter 3. Nonholonomic Variational Problems on Three-Dimensional Lie Groups [51] § 1. The Nonholonomic ε-Sphere and the Wave Front [51] 1.1. Reduction of the Nonholonomic Geodesic Flow [51] 1.2. Metric Tensors on Three-Dimensional Nonholonomic Lie Algebras [52] 1.3. Structure Constants of Three-Dimensional Nonholonomic Lie Algebras [53] 1.4. Normal Forms of Equations of Nonholonomic Geodesics on Three-Dimensional Lie Groups [53] 1.5. The Flow on the Base V + V1 of the Semidirect Product [55] 1.6. Wave Front of Nonholonomic Geodesic Flow, Nonholonomic ε-Sphere and their Singularities [55] 1.7. Metric Structure of the Sphere SeV [58] §2. Nonholonomic Geodesic Flow on Three-Dimensional Lie Groups [61] 2.1. The Monodromy Maps [61] 2.2. Nonholonomic Geodesic Flow on SO(3) [63] 2.3. NG-Flow on Compact Homogeneous Spaces of the Heisenberg Group [65] 2.4. Nonholonomic Geodesic Flows on Compact Homogeneous Spaces of SL2R [67] 2.5. Nonholonomic Geodesic Flow on Some Special Multidimensional Nilmanifolds [70] References [73] Additional Bibliographical Notes [79]
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Traducido de la edición original rusa (Moscú : VINITI, 1987).
Incluye referencias bibliográficas e índices.
Chapter 1. Geometry of Distributions [10] --
§1. Distributions and Related Objects [10] --
1.1. Distributions and Differential Systems [10] --
1.2. Frobenius Theorem and the Flag of a Distribution [12] --
1.3. Codistributions and Pfaffian Systems [14] --
1.4. Regular Distributions [16] --
1.5. Distributions Invariant with Respect to Group Actions and Some Canonical Examples [18] --
1.6. Connections as Distributions [21] --
1.7. A Classification of Left Invariant Contact Structures on Three-Dimensional Lie Groups [22] --
§2. Generic Distributions and Sets of Vector Fields, and Degeneracies of Small Codimension. Nilpotentization and Classification Problem [24] --
2.1. Generic Distributions [25] --
2.2. Normal Forms of Jets of Basic Vector Fields of a Generic Distribution [26] --
2.3. Degeneracies of Small Codimension [27] --
2.4. Generic Sets of Vector Fields [29] --
2.5. Small Codimension Degeneracies of Sets of Vector Fields [31] --
2.6. Projection Map Associated with a Distribution [32] --
2.7. Classificaton of Regular Distributions [33] --
2.8. Nilpotentization and Nilpotent Calculus [34] --
Chapter 2. Basic Theory of Nonholonomic Riemannian Manifolds [35] --
§ 1. General Nonholonomic Variational Problem and the Geodesic Flow on Nonholonomic Riemannian Manifolds [35] --
1.1. Rashevsky-Chow Theorem and Nonholonomic Riemannian Metrics (Camot-Caratheodory Metrics) [35] --
1.2. Two-Point Problem and the Hopf-Rinow Theorem [36] --
1.3. The Cauchy Problem and the Nonholonomic Geodesic Flow [37] --
1.4. The Euler-Lagrange Equations in Invariant Form and in the Orthogonal Moving Frame and Nonholonomic Geodesics [38] --
1.5. The Standard Form of Equations of Nonholonomic Geodesics for Generic Distributions [40] --
1.6. Nonholonomic Exponential Mapping and the Wave Front [41] --
1.7. The Action Functional [42] --
§2. Estimates of the Accessibility Set [42] --
2.1. The Parallelotope Theorem [43] --
2.2. Polysystems and Finslerian Metrics [44] --
2.3. Theorem on the Leading Term [45] --
2.4. Estimates of Generic Nonholonomic Metrics on Compact Manifolds [47] --
2.5. Hausdorff Dimension of Nonholonomic Riemannian Manifolds [48] --
2.6. The Nonholonomic Ball in the Heisenberg Group as the Limit of Powers of a Riemannian Ball [49] --
Chapter 3. Nonholonomic Variational Problems on Three-Dimensional Lie Groups [51] --
§ 1. The Nonholonomic ε-Sphere and the Wave Front [51] --
1.1. Reduction of the Nonholonomic Geodesic Flow [51] --
1.2. Metric Tensors on Three-Dimensional Nonholonomic Lie Algebras [52] --
1.3. Structure Constants of Three-Dimensional Nonholonomic Lie Algebras [53] --
1.4. Normal Forms of Equations of Nonholonomic Geodesics on Three-Dimensional Lie Groups [53] --
1.5. The Flow on the Base V + V1 of the Semidirect Product [55] --
1.6. Wave Front of Nonholonomic Geodesic Flow, Nonholonomic ε-Sphere and their Singularities [55] --
1.7. Metric Structure of the Sphere SeV [58] --
§2. Nonholonomic Geodesic Flow on Three-Dimensional Lie Groups [61] --
2.1. The Monodromy Maps [61] --
2.2. Nonholonomic Geodesic Flow on SO(3) [63] --
2.3. NG-Flow on Compact Homogeneous Spaces of the Heisenberg Group [65] --
2.4. Nonholonomic Geodesic Flows on Compact Homogeneous Spaces of SL2R [67] --
2.5. Nonholonomic Geodesic Flow on Some Special Multidimensional Nilmanifolds [70] --
References [73] --
Additional Bibliographical Notes [79] --
MR, 94h:58069
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