Ants and bracket generating distributions in dimensions 5 and 6. (English) Zbl 1512.70018
A mechanical system of three ants is considered on the floor. Its dynamics is determined by two rules. The first rule (Rule A) equips the 6-dimensional configuration space of the ants with a structure of a homogeneous (3, 6) distribution, while the second rule (Rule B) foliates the configuration space onto 5-dimensional submanifolds consisting of configuration points defining triangles of equal area on the plane. Each of these manifolds is equipped with a homogeneous (2, 3, 5) distribution. The symmetry properties and the local invariants of the velocity distributions of the system are determined in both cases (Rule A and Rule B). Moreover, the singular trajectories (abnormal extremals) of the corresponding distributions are studied in the case of Rule B.
Reviewer: Mikhail I. Krastanov (Sofia)
MSC:
| 70F25 | Nonholonomic systems related to the dynamics of a system of particles |
| 70G65 | Symmetries, Lie group and Lie algebra methods for problems in mechanics |
Keywords:
nonholonomic system; symmetry; linear constraint; abnormal extremal; sub-Riemannian structure; Cartan quarticReferences:
| [1] | Agrachev, A. A., Rolling balls and Octonions, Proceedings of the Steklov Institute of Mathematics, 08/258, 1, 13-22 (2007) · Zbl 1153.70010 |
| [2] | Agrachev, A., Barilari, D., & Boscain, U. (2020). A comprehensive introduction to sub-Riemannian geometry (p. xviii+745). Cambridge. · Zbl 1487.53001 |
| [3] | Agrachev, A. A., & Sachkov, Yu. L. (1999). An Intrinsic Approach to the Control of Rolling Bodies. In Proceedings of the 38th IEEE conference on decision and control, Vol. 1 (pp. 431-435). Phoenix, Arizona, USA. |
| [4] | An, D.; Nurowski, P., Twistor space for rolling bodies, Communications in Mathematical Physics, 326, 393-414 (2014) · Zbl 1296.53100 |
| [5] | An, D.; Nurowski, P., Symmetric \(( 2 , 3 , 5 )\) distributions, an interesting ODE of 7th order and Plebański metric, Journal of Geometry and Physics, 126, 93-100 (2018) · Zbl 1387.34017 |
| [6] | Bateman, H.; Erdelyi, A., Higher transcendental functions, Vol. I (1954), Graw Hill book company: Graw Hill book company N.Y |
| [7] | Bor, G.; Lamoneda, L. H.; Nurowski, P., The dancing metric, \( G_2\)-symmetry and projective rolling, Transactions of the American Mathematical Society, 370, 4433-4481 (2018) · Zbl 1387.53021 |
| [8] | Bor, G.; Montgomery, R., \( G_2\) And the rolling distribution, L’Enseignement Mathematique, 55, 157-196 (2009), arXiv:math/0612469 · Zbl 1251.70008 |
| [9] | Bryant, R., Some aspects of the local and global theory of Pfaffian systems (1979), University of North Carolina at Chapel Hill, (Ph.D. thesis) |
| [10] | Bryant, R., Conformal geometry and 3-plane fields on 6-manifolds, (Developments of cartan geometry and related mathematical problems. Developments of cartan geometry and related mathematical problems, RIMS symposium proceedings, vol. 1502 (2006), Kyoto University), 1-15 |
| [11] | Bryant, R.; Hsu, L., Rigidity of integral curves of rank two distributions, Inventiones Mathematicae, 114, 435-461 (1993) · Zbl 0807.58007 |
| [12] | Čap, A.; Slovak, J., (Parabolic geometries I, Background and general theory. Parabolic geometries I, Background and general theory, Math. surveys monogr., vol. 154 (2009), Amer. Math. Soc) · Zbl 1183.53002 |
| [13] | Cartan, E., Les systemes de Pfaff a cinq variables et les equations aux derivees partielles du seconde ordre, Annales Scientifiques de l’École Normale Supérieure, 27, 109-192 (1910) · JFM 41.0417.01 |
| [14] | Eastwood, M.; Nurowski, P., Aerobatics of flying saucers, Communications in Mathematical Physics, 375, 2335-2365 (2020) · Zbl 1446.53062 |
| [15] | Eastwood, M.; Nurowski, P., Aerodynamics of flying saucers, Communications in Mathematical Physics, 375, 2367-2387 (2020) · Zbl 1440.53040 |
| [16] | Gover, A. R.; Nurowski, P., Obstructions to conformally Einstein metrics in n dimensions, Journal of Geometry and Physics, 56, 450-484 (2006) · Zbl 1098.53014 |
| [17] | Helgason, S., Differential geomtery, Lie groups and symmetric spaces (1978), Academic Press · Zbl 0451.53038 |
| [18] | Hill, C. D.; Nurowski, P., A car as parabolic geometry, (Hervik, S.; Kruglikov, B.; Makina, I.; The, D., Lie theory and applications, Vol. 16 (2022), Springer), 93-130 · Zbl 1509.53013 |
| [19] | Nurowski, P., Differential equations and conformal structures, Journal of Geometry and Physics, 55, 19-49 (2005) · Zbl 1082.53024 |
| [20] | Strazzullo, F., Symmetry analysis of general rank-3 Pfaffian systems in five variables (2009), Utah State University: Utah State University Logan, Utah, (Ph.D. thesis) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
