Geodesic
Liouville integrability of sub-Riemannian problems on Carnot groups of step 4 or greater
Sbornik. Mathematics, Tome 209 (2018) no. 5, pp. 672-713 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

One of the main approaches to investigating sub-Riemannian problems is Mitchell's theorem on nilpotent approximation, which reduces the analysis of a neighbourhood of a regular point to the analysis of the left-invariant sub-Riemannian problem on the corresponding Carnot group. Usually, the in-depth investigation of sub-Riemannian shortest paths is based on integrating the Hamiltonian system of Pontryagin's maximum principle explicitly. We give new formulae for sub-Riemannian geodesics on a Carnot group with growth vector $(2,3,5,6)$ and prove that left-invariant sub-Riemannian problems on free Carnot groups of step 4 or greater are Liouville nonintegrable. Bibliography: 30 titles.
Keywords: sub-Riemannian geometry, Liouville integrability, Carnot groups, growth vector, separatrix splitting
Mots-clés : Melnikov-Poincaré method. 
@article{SM_2018_209_5_a3,
     author = {L. V. Lokutsievskiy and Yu. L. Sachkov},
     title = {Liouville integrability of {sub-Riemannian} problems on {Carnot} groups of step 4 or greater},
     journal = {Sbornik. Mathematics},
     pages = {672--713},
     year = {2018},
     volume = {209},
     number = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2018_209_5_a3/}
}
TY  - JOUR
AU  - L. V. Lokutsievskiy
AU  - Yu. L. Sachkov
TI  - Liouville integrability of sub-Riemannian problems on Carnot groups of step 4 or greater
JO  - Sbornik. Mathematics
PY  - 2018
SP  - 672
EP  - 713
VL  - 209
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/SM_2018_209_5_a3/
LA  - en
ID  - SM_2018_209_5_a3
ER  - 
%0 Journal Article
%A L. V. Lokutsievskiy
%A Yu. L. Sachkov
%T Liouville integrability of sub-Riemannian problems on Carnot groups of step 4 or greater
%J Sbornik. Mathematics
%D 2018
%P 672-713
%V 209
%N 5
%U http://geodesic.mathdoc.fr/item/SM_2018_209_5_a3/
%G en
%F SM_2018_209_5_a3
L. V. Lokutsievskiy; Yu. L. Sachkov. Liouville integrability of sub-Riemannian problems on Carnot groups of step 4 or greater. Sbornik. Mathematics, Tome 209 (2018) no. 5, pp. 672-713. http://geodesic.mathdoc.fr/item/SM_2018_209_5_a3/

[1] Yu. L. Sachkov, E. F. Sachkova, “Degenerate abnormal trajectories in a sub-Riemannian problem with growth vector $(2,3,5,8)$”, Differ. Equ., 53:3 (2017), 352–365 | DOI | DOI | MR | Zbl

[2] A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monogr. Math., Springer, Berlin, 2007, xxvi+800 pp. | DOI | MR | Zbl

[3] J.-P. Gauthier, Yu. L. Sachkov, “On the free Carnot $(2, 3, 5, 8)$ group”, Programmnye sistemy: teoriya i prilozheniya, 6:2 (2015), 45–61 | DOI

[4] J. Mitchell, “On Carnot–Carathéodory metrics”, J. Differential Geom., 21:1 (1985), 35–45 | DOI | MR | Zbl

[5] A. Bellaïche, “The tangent space in sub-Riemannian geometry”, Sub-Riemannian geometry, Progr. Math., 144, Birkhäuser, Basel, 1996, 1–78 | DOI | MR | Zbl

[6] M. Gromov, “Carnot–Carathéodory spaces seen from within”, Sub-Riemannian geometry, Progr. Math., 144, Birkhäuser, Basel, 1996, 79–323 | DOI | MR | Zbl

[7] R. W. Brockett, “Control theory and singular Riemannian geometry”, New directions in applied mathematics (Cleveland, OH, 1980), Springer, New York–Berlin, 1982, 11–27 | DOI | MR | Zbl

[8] A. M. Vershik, V. Y. Gershkovich, “Nonholonomic dynamical systems, geometry of distributions and variational problems”, Dynamical systems VII, Encyclopaedia Math. Sci., 16, Springer-Verlag, Berlin, 1994, 1–81 | DOI | MR | MR | Zbl

[9] R. W. Brockett, Liyi Dai, “Non-holonomic kinematics and the role of elliptic functions in constructive controllability”, Nonholonomic motion planning, Kluwer Acad. Publ., Dordrecht, 1993, 1–21 | DOI | Zbl

[10] Yu. L. Sachkov, “Exponential map in the generalized Dido problem”, Sb. Math., 194:9 (2003), 1331–1359 | DOI | DOI | MR | Zbl

[11] Yu. L. Sachkov, “Discrete symmetries in the generalized Dido problem”, Sb. Math., 197:2 (2006), 235–257 | DOI | DOI | MR | Zbl

[12] Yu. L. Sachkov, “The Maxwell set in the generalized Dido problem”, Sb. Math., 197:4 (2006), 595–621 | DOI | DOI | MR | Zbl

[13] Yu. L. Sachkov, “Complete description of the Maxwell strata in the generalized Dido problem”, Sb. Math., 197:6 (2006), 901–950 | DOI | DOI | MR | Zbl

[14] A. A. Agrachev, Yu. L. Sachkov, Control theory from the geometric viewpoint, Encyclopaedia Math. Sci., 87, Control theory and optimization, II, Springer-Verlag, Berlin, 2004, xiv+412 pp. | DOI | MR | Zbl | Zbl

[15] A. A. Kirillov, Lectures on the orbit method, Grad. Stud. Math., 64, Amer. Math. Soc., Providence, RI, 2004, xx+408 pp. | DOI | MR | Zbl

[16] A. V. Bolsinov, A. T. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, Chapman Hall/CRC, Boca Raton, FL, 2004, xvi+730 pp. | DOI | MR | MR | Zbl | Zbl

[17] B. Gaveau, “Principe de moindre action, propagation de la chaleur et estimees sous elliptiques sur certains groupes nilpotents”, Acta Math., 139:1-2 (1977), 95–153 | DOI | MR | Zbl

[18] O. Myasnichenko, “Nilpotent $(n,n(n+1)/2)$ sub-Riemannian problem”, J. Dynam. Control Systems, 12:1 (2002), 87–95 | DOI | MR | Zbl

[19] A. Anzaldo-Meneses, F. Monroy-Pérez, “Goursat distribution and sub-Riemannian structures”, J. Math. Phys., 44:12 (2003), 6101–6111 | DOI | MR | Zbl

[20] L. V. Lokutsievskiy, Yu. L. Sachkov, “Non-integrability of geodesic flows in left-invariant subriemannian problems of step 4”, Metric Structures and Control Systems (Novosibirsk, 2015), Sobolev institute of mathematics, Novosibirsk, 2015, 26

[21] A. A. Kirillov, “Local Lie algebras”, Russian Math. Surveys, 31:4 (1976), 55–75 | DOI | MR | Zbl

[22] V. V. Kozlov, D. V. Treschev, “Topology of the configuration space, singularities of the potential, and polynomial integrals of equations of dynamics”, Sb. Math., 207:10 (2016), 1435–1449 | DOI | DOI | MR | Zbl

[23] V. V. Kozlov, “Rational integrals of quasi-homogeneous dynamical systems”, J. Appl. Math. Mech., 79:3 (2015), 209–216 | DOI | MR

[24] A. E. H. Love, A treatise on the mathematical theory of elasticity, 4th ed., Cambridge Univ. Press, Cambridge, 1927, xviii+643 pp. | MR | Zbl

[25] N. Boizot, J.-P. Gauthier, “On the motion planning of the ball with a trailer”, Math. Control Relat. Fields, 3:3 (2013), 269–286 | DOI | MR | Zbl

[26] G. A. Korn, T. M. Korn, Mathematical handbook for scientists and engineers, 2nd ed., McGraw-Hill Book Co., New York–Toronto, ON–London, 1968, xvii+1130 pp. | MR | MR | Zbl | Zbl

[27] D. Treschev, O. Zubelevich, Introduction to the perturbation theory of Hamiltonian systems, Springer Monogr. Math., Springer-Verlag, Berlin, 2010, x+211 pp. | DOI | MR | Zbl

[28] S. L. Ziglin, “Splitting of the separatrices and the nonexistence of first integrals in systems of differential equations of Hamiltonian type with two degrees of freedom”, Math. USSR-Izv., 31:2 (1988), 407–421 | DOI | MR | Zbl

[29] E. Zehnder, “Homoclinic points near elliptic fixed points”, Comm. Pure Appl. Math., 26:2 (1973), 131–182 | DOI | MR | Zbl

[30] E. T. Whittaker, G. N. Watson, A course of modern analysis, 4th ed., reprinted, Cambridge Univ. Press, Cambridge, 1950, vii+608 pp. | MR | Zbl | Zbl