Abstract
In this paper we study the nilpotent (3, 6) sub-Riemannian problem. We describe the envelope of sub-Riemannian geodesics starting from a fixed point. We also describe the wave fronts propagating from the point. For general nilpotent (n,n(n + 1)/2) sub-Riemannian problem we formulate a conjecture about the form of the variety where geodesics starting from a fixed point lose optimality.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Bellaiche, The tangent space in sub-Riemannian geometry. In: Sub-Riemannian Geometry (A. Bellaiche and J.-J. Risler (Eds.)), Birkhäuser, 1996.
M. Gromov, Carnot-Carateodory spaces seen from within. In: Sub-Riemannian Geometry (A. Bellaiche and J.-J. Risler (Eds.)), Birkhäuser, 1996.
R. W. Brockett, Control theory and singular Riemannian geometry. In: New Directions in Applied Mathematics (P. J. Hilton and G. S. Young (Eds.)), Springer-Verlag, 1981.
A. M. Vershik and V. Y. Gershkovich, Nonholonomic dynamical systems. Geometry of distributions and variational problems. (Russian) In: Itogi Nauki i Tekhniki, Sovrem. Probl. Mat. Fund. Naprav. 16, VINITI, Moscow (1987), 5–85. (English translation in: Encyclopaedia Math. Sci. 16, Dynamical Systems 7, Springer-Verlag.)
A. A. Agrachev, Exponential mappings for contact sub-Riemannian structures, J. Dynam. Control Systems 2 (1996), No. 3, 321–358.
V. I. Arnold, Mathematical methods in classical mechanics, Nauka, Moscow, 1989.
A. A. Agrachev and A. V. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity, Ann. Inst. H. Poincaré Anal. Non linéaire, 13 (1996) 635–690.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Myasnichenko, O. Nilpotent (3, 6) Sub-Riemannian Problem. Journal of Dynamical and Control Systems 8, 573–597 (2002). https://doi.org/10.1023/A:1020719503741
Issue Date:
DOI: https://doi.org/10.1023/A:1020719503741