Contact hamiltonians distinguishing locally certain Goursat systems

Piotr Mormul

Banach Center Publications (2000)

  • Volume: 51, Issue: 1, page 219-230
  • ISSN: 0137-6934

Abstract

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For the first time in dimension 9, the Goursat distributions are not locally smoothly classified by their small growth vector at a point. As shown in [M1], in dimension 9 of the underlying manifold 93 different local behaviours are possible and four irregular pairs of them have coinciding small growth vectors. In the present paper we distinguish geometrically objects in three of those pairs. Smooth functions in three variables - contact hamiltonians in the terminology of Arnold, [A] - help to do that. One pair of models, however, resists this technique. Another example of similar resistance in dimension 10 is also given - through the exact classification in dimension 10 of one family of local pseudo-normal forms (with redundant real constants) for Goursat objects. The latter result is an harbinger of more general phenomena that will be treated in a subsequent paper.

How to cite

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Mormul, Piotr. "Contact hamiltonians distinguishing locally certain Goursat systems." Banach Center Publications 51.1 (2000): 219-230. <http://eudml.org/doc/209034>.

@article{Mormul2000,
abstract = {For the first time in dimension 9, the Goursat distributions are not locally smoothly classified by their small growth vector at a point. As shown in [M1], in dimension 9 of the underlying manifold 93 different local behaviours are possible and four irregular pairs of them have coinciding small growth vectors. In the present paper we distinguish geometrically objects in three of those pairs. Smooth functions in three variables - contact hamiltonians in the terminology of Arnold, [A] - help to do that. One pair of models, however, resists this technique. Another example of similar resistance in dimension 10 is also given - through the exact classification in dimension 10 of one family of local pseudo-normal forms (with redundant real constants) for Goursat objects. The latter result is an harbinger of more general phenomena that will be treated in a subsequent paper.},
author = {Mormul, Piotr},
journal = {Banach Center Publications},
keywords = {Goursat systems; contact Hamiltonians},
language = {eng},
number = {1},
pages = {219-230},
title = {Contact hamiltonians distinguishing locally certain Goursat systems},
url = {http://eudml.org/doc/209034},
volume = {51},
year = {2000},
}

TY - JOUR
AU - Mormul, Piotr
TI - Contact hamiltonians distinguishing locally certain Goursat systems
JO - Banach Center Publications
PY - 2000
VL - 51
IS - 1
SP - 219
EP - 230
AB - For the first time in dimension 9, the Goursat distributions are not locally smoothly classified by their small growth vector at a point. As shown in [M1], in dimension 9 of the underlying manifold 93 different local behaviours are possible and four irregular pairs of them have coinciding small growth vectors. In the present paper we distinguish geometrically objects in three of those pairs. Smooth functions in three variables - contact hamiltonians in the terminology of Arnold, [A] - help to do that. One pair of models, however, resists this technique. Another example of similar resistance in dimension 10 is also given - through the exact classification in dimension 10 of one family of local pseudo-normal forms (with redundant real constants) for Goursat objects. The latter result is an harbinger of more general phenomena that will be treated in a subsequent paper.
LA - eng
KW - Goursat systems; contact Hamiltonians
UR - http://eudml.org/doc/209034
ER -

References

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  1. [A] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer 1978. Zbl0386.70001
  2. [C] E. Cartan, Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes, Bull. Soc. math. France 42 (1914), 1-36. Zbl45.0472.04
  3. [CM] M. Cheaito and P. Mormul, Rank-2 distributions satisfying the Goursat condition: all their local models in dimension 7 and 8, ESAIM: Control, Optimisation and Calculus of Variations (URL: http:/www.emath.fr/cocv/) 4 (1999), 137-158. Zbl0957.58002
  4. [G] M. Gaspar, Sobre la clasificacion de sistemas de Pfaff en bandera, in: Proceedings of 10th Spanish-Portuguese Conference on Math., University of Murcia (1985), 67-74 (in Spanish). 
  5. [K] A. Kumpera, handwritten notes, April 1998. 
  6. [KR] A. Kumpera, C. Ruiz, Sur l'équivalence locale des systèmes de Pfaff en drapeau, in: F. Gherardelli ed., Monge-Ampère Equations and Related Topics, Inst.Alta Mat., Rome 1982, 201-248. Zbl0516.58004
  7. [Li] S. Lie, Geometrie der Berührungstransformationen, and other related papers, Gesamm. Abh. Vol.III, 221-251; Vol.IV, 1-96, 265-290. 
  8. [Ly] V. Lychagin, Local classification of non-linear first order partial differential equations, Russian Math.Surveys 30 (1975), 105-175 (English translation). 
  9. [M1] P. Mormul, Local classification of rank-2 distributions satisfying the Goursat condition in dimension 9, preprint N°582 Inst.of Math., Polish Acad. Sci. (1998); submitted to the Proceedings of the conference 'Singularités et Géométrie Sous-Riemannienne', Chambéry, October 1997. 
  10. [M2] P. Mormul, Goursat distributions with one singular hypersurface - constants important in their Kumpera-Ruiz pseudo-normal forms, preprint N°185, Labo de Topologie, Univ. de Bourgogne, Dijon (1999). 
  11. [PR] W. Pasillas-Lépine and W. Respondek, On the geometry of Goursat structures, preprint (1999). Zbl0966.58002

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