On the free Carnot (2, 3, 5, 8) group
Автор: Gauthier Jean-Paul, Sachkov Yuri Leonidovich
Журнал: Программные системы: теория и приложения @programmnye-sistemy
Рубрика: Методы оптимизации и теория управления
Статья в выпуске: 2 (25) т.6, 2015 года.
Бесплатный доступ
We consider the free nilpotent Lie algebra with 2 generators, of step 4, and the corresponding connected simply connected Lie group 𝐺, with the aim to study the left-invariant sub-Riemannian structure on defined by the generators of as an orthonormal frame. We compute two vector field models of by polynomial vector fields in R8, and find an infinitesimal symmetry of the sub-Riemannian structure. Further, we compute explicitly the product rule in and the right-invariant frame on 𝐺.
Carnot group., sub-riemannian geometry
Короткий адрес: https://sciup.org/14336030
IDR: 14336030
Список литературы On the free Carnot (2, 3, 5, 8) group
- M. Gromov. Carnot-Caratheodory spaces seen from within//SubRiemannian geometry, Progress in Mathematics, vol. 144, Birkhauser Basel, 1996. P. 79-323.
- J. Mitchell. On Carnot-Caratheodory metrics//J. Differential Geom., 21 (1985). P. 35-45.
- A. BellA¨ ıche. The tAngent spAce in sub-RiemAnniAn geometry // Sub-RiemAnniAn geometry, Progress in MAthemAtics, vol. 144, BirkhAuser BAsel, 1996. P. 1-78.
- A. A. Agrachev, A. A. Sarychev. Filtration of a Lie algebra of vector fields and nilpotent approximation of control systems//Dokl. Akad. Nauk SSSR, 295 (1987). P. 104-108.
- R. Montgomery. A tour of sub-Riemannian geometries, their geodesics and applications: control theory from the geometric viewpoint, American Mathematical Society, 2002.
- R. Brockett. Control theory and singular Riemannian geometry//New directions in applied mathematics, Springer-Verlag, New York, 1982. P. 11-27.
- A. M. Vershik, V. Y. Gershkovich. Nonholonomic dynamical systems. Geometry of distributions and variational problems//Dynamical systems -7, Itogi Nauki i Tekhniki: Sovremennye Problemy Matematiki, Fundamental’nyje Napravleniya, vol. 16, VINITI, M., 1987. P. 5-85.
- Yu. L. Sachkov. Exponential mapping in generalized Dido’s problem//Mat. Sbornik, V. 194. No. 9. (2003). P. 63-90.
- Yu. L. Sachkov. Discrete symmetries in the generalized Dido problem//Matem. Sbornik, V. 197. No. 2. (2006). P. 95-116.
- Yu. L. Sachkov. The Maxwell set in the generalized Dido problem//Matem. Sbornik, V. 197. No. 4. (2006). P. 123-150.
- Yu. L. Sachkov. Complete description of the Maxwell strata in the generalized Dido problem//Matem. Sbornik, V. 197. No. 6. (2006). P. 111-160.
- R. Monti. The regularity problem for sub-Riemannian geodesics//Geometric Control Theory and sub-Riemannian Geometry, Springer INdAM Series, vol. 5, Springer International Publishing, 2014. P. 313-332.
- M. Grayson, R. Grossman. Vector fields and nilpotent Lie algebras//Symbolic Computation: Applications to Scientific Computing, Frontiers in Applied Mathematics, ed. R. Grossman, 1989. P. 77-96.
- Yu. Sachkov. On Carnot algebra with the growth vector (2, 3, 5, 8), http://arxiv.org/abs/1304.1035v1,2013, 13 p.
- Ch. Reutenauer, Free Lie algebras, London Mathematical Society Monographs New Series, vol. 7, Oxford University Press, 1993.
- M. Hall. A basis for free Lie rings and higher commutators in free groups//Proc. Amer. Math. Soc., 1 (1950). P. 575-581.
- A. A. Agrachev, Yu. L. Sachkov. Control theory from the geometric viewpoint, Springer-Verlag, Berlin, 2004.
