Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Introduction
- 1 Geometry of Surfaces in R3
- 2 Vector Fields
- 3 Sub-Riemannian Structures
- 4 Pontryagin Extremals: Characterization and Local Minimality
- 5 First Integrals and Integrable Systems
- 6 Chronological Calculus
- 7 Lie Groups and Left-Invariant Sub-Riemannian Structures
- 8 Endpoint Map and Exponential Map
- 9 2D Almost-Riemannian Structures
- 10 Nonholonomic Tangent Space
- 11 Regularity of the Sub-Riemannian Distance
- 12 Abnormal Extremals and Second Variation
- 13 Some Model Spaces
- 14 Curves in the Lagrange Grassmannian
- 15 Jacobi Curves
- 16 Riemannian Curvature
- 17 Curvature in 3D Contact Sub-Riemannian Geometry
- 18 Integrability of the Sub-Riemannian Geodesic Flow on 3D Lie Groups
- 19 Asymptotic Expansion of the 3D Contact Exponential Map
- 20 Volumes in Sub-Riemannian Geometry
- 21 The Sub-Riemannian Heat Equation
- Appendix Geometry of Parametrized Curves in Lagrangian Grassmannians Igor Zelenko
- References
- Index
18 - Integrability of the Sub-Riemannian Geodesic Flowon 3D Lie Groups
Published online by Cambridge University Press: 28 October 2019
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Introduction
- 1 Geometry of Surfaces in R3
- 2 Vector Fields
- 3 Sub-Riemannian Structures
- 4 Pontryagin Extremals: Characterization and Local Minimality
- 5 First Integrals and Integrable Systems
- 6 Chronological Calculus
- 7 Lie Groups and Left-Invariant Sub-Riemannian Structures
- 8 Endpoint Map and Exponential Map
- 9 2D Almost-Riemannian Structures
- 10 Nonholonomic Tangent Space
- 11 Regularity of the Sub-Riemannian Distance
- 12 Abnormal Extremals and Second Variation
- 13 Some Model Spaces
- 14 Curves in the Lagrange Grassmannian
- 15 Jacobi Curves
- 16 Riemannian Curvature
- 17 Curvature in 3D Contact Sub-Riemannian Geometry
- 18 Integrability of the Sub-Riemannian Geodesic Flow on 3D Lie Groups
- 19 Asymptotic Expansion of the 3D Contact Exponential Map
- 20 Volumes in Sub-Riemannian Geometry
- 21 The Sub-Riemannian Heat Equation
- Appendix Geometry of Parametrized Curves in Lagrangian Grassmannians Igor Zelenko
- References
- Index
Summary
In this chapter we show how to find certain firstintegrals, for Hamiltonian systems on Lie groups,that are automatically in involution with each otherand with the Hamiltonian. This theory will be usedto prove that the Hamiltonian system for normalPontryagin extremals for rank-2 left-invariantsub-Riemannian structures on three-dimensional Liegroups is completely integrable.
- Type
- Chapter
- Information
- A Comprehensive Introduction to Sub-Riemannian Geometry , pp. 607 - 632Publisher: Cambridge University PressPrint publication year: 2019