Abstract
It is shown how to produce new examples of integrable Hamiltonian dynamical systems of differential geometric origin. These are normal geodesic flows of homogeneous Carnot-Carathéodory metrics. The relation to previous descriptions of such flows via non-Hamiltonian methods and to problems of analytic mechanics is discussed.
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References
V.I. Arnol'd, Mathematical methods of classical mechanics.,Springer-Verlag, Berlin-Heidelberg-New York, 1978.
V.I. Arnol'd, V.V. Kozlov, and A.I. Nejshtadt, Mathematical aspects of classical and celestial mechanics. In: Dynamical Systems III, Encyclopaedia of Mathematical Sciences, Vol. 3,Springer-Verlag, Berlin-Heidelberg-New York, 1988.
V.N. Berestovskii, Homogeneous spaces with intrinsic metric.Sov. Math. Dokl. 38 (1989), 60–63.
—, Geodesics of nonholonomic left-invariant intrinsic metrics on the Heisenberg group and isoperimetric curves on the Minkowskii plane. (Russian)Siberian Math. J. 35 (1994), 1–8.
L. Cesari, Optimization theory and applications.Springer-Verlag, New York, 1983.
S.A. Chaplygin, Investigations in the dynamics of nonholonomic systems. (Russian)Gostekhizdat, Moscow-Leningrad, 1947.
M. Gromov, Carnot-Carathéodory spaces seeing from within.Preprint, IHES, 1994.
U. Hamenstädt, Some regularity theorems for Carnot-Carathéodory metrics.J. Differ. Geom 32 (1990), 819–850.
J. Mitchell, On Carnot-Carathéodory metrics.J. Differ. Geom. 21 (1985), 35–45.
R. Montgomery, Abnormal minimizers.SIAM J. Control and Optimiz. 32 (1994), 1605–1620.
S.P. Novikov and I. Shmel'tser, Periodic solutions of the Kirchhoff equations for the free motion of a rigid body in a liquid, and the extended Lyusternik-Schnirelmann-Morse (LSM) theory. I.Funct. Anal. and Appl. 15 (1981), 197–207.
G. Paternain, On the topology of manifolds with completely integrable geodesic flows.Ergodic Theory and Dynam. Syst. 12 (1992), 109–121.
R.S. Strichartz, Sub-Riemannian geometry,J. Differ. Geom. 24 (1986), 221–263;J. Differ. Geom. 30 (1989), 595–596.
I.A. Taimanov, Topological obstructions to integrability of geodesic flows on non-simply-connected manifolds.Math. USSR Izv. 30 (1988), 403–409.
—, The topology of Riemannian manifolds with integrable geodesic flows.Proc. Steklov Inst. Math. 205 (1995), 139–150.
T.J.S. Taylor, Some aspects of differential geometry associated with hypoelliptic second order operators.Pacif. J. Math. 136 (1989), 355–378.
A.M. Vershik and V.Ya. Gershkovich, Nonholonomic dynamical systems, geometry of distributions, and variational problems. In: Dynamical Systems VII, Encyclopaedia of Mathematical Sciences, Vol. 16,Springer-Verlag, Berlin-Heidelberg-New York, 1994, 1–81.
A.P. Veselov and L.E. Veselova, Integrable nonholonomic systems on Lie groups. (Russian)Math. Notes 44 (1988), 810–819.
S.K. Vodopjanov and A.V. Greshnov, On extension of functions with bounded mean oscillation onto a space of homogeneous type with intrinsic metric. (Russian)Siberian Math. J. 37 (1996).
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Taimanov, I.A. Integrable geodesic flows of nonholonomic metrics. Journal of Dynamical and Control Systems 3, 129–147 (1997). https://doi.org/10.1007/BF02471765
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DOI: https://doi.org/10.1007/BF02471765