Abstract
The coadjoint orbits and the Casimir functions are described for free nilpotent Lie groups of step \(2\). The symplectic foliation consists of affine subspaces in the Lie coalgebra. Left-invariant time-optimal problems are considered on Carnot groups of step \(2\) for which the set of admissible velocities is a strictly convex compact set in the first layer of the Lie algebra that contains the origin in its interior. The first integrals of the vertical subsystem of the Hamiltonian system of the Pontryagin maximum principle are described. For two-dimensional coadjoint orbits, the constancy and periodicity properties of the solutions of this subsystem, as well as the phase flow, are described.
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References
R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications (Amer. Math. Soc., Providence, RI, 2002).
A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint (Springer, Berlin, 2004).
A. Agrachev, D. Barilari, and U. Boscain, A Comprehensive Introduction to Sub-Riemannian Geometry (Cambridge Univ. Press, Cambridge, 2019).
V. N. Berestovskii, “Homogeneous manifolds with intrinsic metric. II,” Sib. Math. J. 30 (2), 180–191 (1989).
V. N. Berestovskii, “Geodesics of nonholonomic left-invariant intrinsic metrics on the Heisenberg group and isoperimetric curves on the Minkowski plane,” Sib. Math. J. 35 (1), 3–11 (1994).
D. Barilari, U. Boscain, E. Le Donne, and M. Sigalotti, “Sub-Finsler structures from the time-optimal control viewpoint for some nilpotent distributions,” J. Dyn. Control Syst. 23 (3), 547–575 (2017).
A. A. Ardentov, E. Le Donne, and Yu. L. Sachkov, “Sub-Finsler Geodesics on the Cartan Group,” Regul. Chaotic Dyn. 24 (1), 36–60 (2019).
E. Le Donne and G. Speight, “Lusin approximation for horizontal curves in step 2 Carnot groups,” Calc. Var. Partial Differential Equations 55 (5, Art. 111) (2016).
E. Le Donne and S. Rigot, “Remarks about Besicovitch covering property in Carnot groups of step 3 and higher,” Proc. Amer. Math. Soc. 144 (5), 2003–2013 (2016).
Yu. Sachkov, “Periodic controls in step 2 strictly convex sub-Finsler problems,” Regul. Chaotic Dyn. 25 (1), 33–39 (2020).
H. Busemann, “The isoperimetric problem in the Minkowski plane,” Amer. J. Math. 69 (4), 863–871 (1947).
R. W. Brockett, “Nonlinear control theory and differential geometry,” in Proceedings of the International Congress of Mathematicians (PWN, Warszawa, 1983), pp. 1357–1368.
O. Myasnichenko, “Nilpotent \((3,6)\) sub-Riemannian problem,” J. Dynam. Control Systems 8 (4), 573–597 (2002).
L. Rizzi and U. Serres, “On the cut locus of free, step two Carnot groups,” Proc. Amer. Math. Soc. 145 (12), 5341–5357 (2017).
A. A. Kirillov, Lectures on the Orbit Method (Amer. Math. Soc., Providence, RI, 2004).
R. T. Rockafellar, Convex Analysis (Princeton Univ. Press, Princeton, NJ, 1970).
V. Jurdjevic, Geometric Control Theory (Cambridge Univ. Press, Cambridge, 1997).
E. Hakavuori, Infinite Geodesics and Isometric Embeddings in Carnot Groups of Step 2, arXiv: 1905.03214 (2019).
Yu. L. Sachkov, “Exponential map in the generalized Dido problem,” Sb. Math. 194 (9), 1331–1359 (2003).
I. A. Bizyaev, A. V. Borisov, A. A. Kilin, and I. S. Mamaev, “Integrability and nonintegrability of sub-Riemannian geodesic flows on Carnot groups,” Regul. Chaotic Dyn. 21 (6), 759–774 (2016).
L. V. Lokutsievskii and Yu. L. Sachkov, “Liouville integrability of sub-Riemannian problems on Carnot groups of step 4 or greater,” Sb. Math. 209 (5), 672–713 (2018).
A. V. Podobryaev, “Casimir functions of free nilpotent Lie groups of steps three and four,” J. Dyn. Control Syst. (in press); arXiv: 2006.00224 (2020).
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This work was supported by the Russian Science Foundation under grant 17-11-01387-P at Ailamazyan Program Systems Institute of Russian Academy of Sciences.
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Sachkov, Y.L. Coadjoint Orbits and Time-Optimal Problems for Step-\(2\) Free Nilpotent Lie Groups. Math Notes 108, 867–876 (2020). https://doi.org/10.1134/S0001434620110280
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DOI: https://doi.org/10.1134/S0001434620110280