Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 10 (2014), 059, 38 pages      arXiv:1205.2992      https://doi.org/10.3842/SIGMA.2014.059

Configurations of an Articulated Arm and Singularities of Special Multi-Flags

Fernand Pelletier a and Mayada Slayman b
a) Université de Savoie, Laboratoire de Mathématiques (LAMA), Campus Scientifique, 73376 Le Bourget-du-Lac Cedex, France
b) Department of Mathematical Sciences, Faculty of Sciences II, Lebanese University, Lebanon

Received January 29, 2013, in final form May 18, 2014; Published online June 05, 2014

Abstract
P. Mormul has classified the singularities of special multi-flags in terms of “EKR class'' encoded by sequences $j_1,\dots, j_k$ of integers (see [Singularity Theory Seminar, Warsaw University of Technology, Vol. 8, 2003, 87-100] and [Banach Center Publ., Vol. 65, Polish Acad. Sci., Warsaw, 2004, 157-178]). However, A.L. Castro and R. Montgomery have proposed in [Israel J. Math. 192 (2012), 381-427] a codification of singularities of multi-flags by RC and RVT codes. The main results of this paper describe a decomposition of each ''EKR'' set of depth $1$ in terms of RVT codes as well as characterize such a set in terms of configurations of an articulated arm. Indeed, an analogue description for some ''EKR'' sets of depth $2$ is provided. All these results give rise to a complete characterization of all ''EKR'' sets for $1\leq k\leq 4$.

Key words: special multi-flags distributions; Cartan prolongation; spherical prolongation; articulated arm; rigid bar.

pdf (639 kb)   tex (70 kb)

References

  1. Adachi J., Global stability of distributions of higher corank of derived length one, Int. Math. Res. Not. 2003 (2003), 2621-2638.
  2. Adachi J., Global stability of special multi-flags, Israel J. Math. 179 (2010), 29-56.
  3. Castro A.L., Howard W.C., A Monster Tower approach to Goursat multi-flags, Differential Geom. Appl. 30 (2012), 405-427.
  4. Castro A.L., Montgomery R., Spatial curve singularities and the Monster/Semple tower, Israel J. Math. 192 (2012), 381-427.
  5. Kumpera A., Rubin J.L., Multi-flag systems and ordinary differential equations, Nagoya Math. J. 166 (2002), 1-27.
  6. Li S.J., Respondek W., The geometry, controllability, and flatness property of the $n$-bar system, Internat. J. Control 84 (2011), 834-850.
  7. Montgomery R., Zhitomirskii M., Geometric approach to Goursat flags, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), 459-493.
  8. Mormul P., Geometric singularity classes for special $k$-flags, $k \geq 2$, of arbitrary length, in Singularity Theory Seminar, Editor S. Janeczko, Warsaw University of Technology, Vol. 8, 2003, 87-100.
  9. Mormul P., Multi-dimensional Cartan prolongation and special $k$-flags, in Geometric Singularity Theory, Banach Center Publ., Vol. 65, Polish Acad. Sci., Warsaw, 2004, 157-178.
  10. Mormul P., Pelletier F., Special 2-flags in lengths not exceeding four: a study in strong nilpotency of distributions, arXiv:1011.1763.
  11. Pasillas-Lépine W., Respondek W., Contact systems and corank one involutive subdistributions, Acta Appl. Math. 69 (2001), 105-128, math.DG/0004124.
  12. Pasillas-Lépine W., Respondek W., On the geometry of Goursat structures, ESAIM Control Optim. Calc. Var. 6 (2001), 119-181, math.DG/9911101.
  13. Pelletier F., Espace de configuration d'un système mécanique et tours de fibrés associées à un multi-drapeau spécial, C. R. Math. Acad. Sci. Paris 350 (2012), 71-76.
  14. Shibuya K., Yamaguchi K., Drapeau theorem for differential systems, Differential Geom. Appl. 27 (2009), 793-808.
  15. Slayman M., Bras articulé et distributions multi-drapeaux, Ph.D. Thesis, Université de Savoie, Laboratoire de Mathématiques (LAMA), 2008.
  16. Slayman M., Pelletier F., Articulated arm and special multi-flags, J. Math. Sci. Adv. Appl. 8 (2011), 9-41, arXiv:1205.2990.


Previous article  Next article   Contents of Volume 10 (2014)