Geodesic
The geometry of integrable and superintegrable systems
Teoretičeskaâ i matematičeskaâ fizika, Tome 172 (2012) no. 2, pp. 264-274 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider the automorphism group of the geometry of an integrable system. The geometric structure used to obtain it is generated by a normal-form representation of integrable systems that is independent of any additional geometric structure like symplectic, Poisson, etc. Such a geometric structure ensures a generalized toroidal bundle on the carrier space of the system. Noncanonical diffeomorphisms of this structure generate alternative Hamiltonian structures for completely integrable Hamiltonian systems. The energy–period theorem for dynamical systems implies the first nontrivial obstruction to the equivalence of integrable systems.
Keywords: integrable system, superintegrable system, energy–period theorem, geometric structure. 
@article{TMF_2012_172_2_a6,
     author = {A. Ibort and G. Marmo},
     title = {The~geometry of integrable and superintegrable systems},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {264--274},
     year = {2012},
     volume = {172},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2012_172_2_a6/}
}
TY  - JOUR
AU  - A. Ibort
AU  - G. Marmo
TI  - The geometry of integrable and superintegrable systems
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2012
SP  - 264
EP  - 274
VL  - 172
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2012_172_2_a6/
LA  - ru
ID  - TMF_2012_172_2_a6
ER  - 
%0 Journal Article
%A A. Ibort
%A G. Marmo
%T The geometry of integrable and superintegrable systems
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2012
%P 264-274
%V 172
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2012_172_2_a6/
%G ru
%F TMF_2012_172_2_a6
A. Ibort; G. Marmo. The geometry of integrable and superintegrable systems. Teoretičeskaâ i matematičeskaâ fizika, Tome 172 (2012) no. 2, pp. 264-274. http://geodesic.mathdoc.fr/item/TMF_2012_172_2_a6/

[1] F. Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen, Erlangen Program, Univ. Erlangen, Erlangen, 1872 | Zbl

[2] F. Klein, Bull. Amer. Math. Soc., 2 (1893), 215–249 | DOI | MR | Zbl

[3] F. Klein, A comparative review of recent researches in geometry, arXiv: 0807.3161 | MR

[4] A. Banyaga, Proc. Amer. Math. Soc., 98:1 (1986), 113–118 | DOI | MR | Zbl

[5] A. Banyaga, J. Differ. Geom., 28:1 (1988), 23–35 | DOI | MR | Zbl

[6] J. Grabowski, “Isomorphisms of Poisson and Jacobi brackets”, Poisson Geometry, Stanislaw Zakrzewski in Memoriam (Warsaw, Poland, August 3–15, 1998), Banach Center Publ., 51, eds. J. Grabowski, P. Urbański, Polish Acad. Sci., Warsaw, 2000, 79–85 | MR | Zbl

[7] S. de Filippo, G. Landi, G. Marmo, G. Vilasi, Ann. Inst. H. Poincaré, 50:2 (1989), 205–218 | MR | Zbl

[8] T. Levi-Civita, J. Math. Phys., 13 (1934), 18–40 | DOI | Zbl

[9] M. Giordano, G. Marmo, C. Rubano, Inverse Problems, 9:4 (1993), 443–467 | DOI | MR | Zbl

[10] E. Ercolessi, G. Marmo, G. Morandi, Riv. Nuovo Cim., 33:8–9 (2010), 401–590 | DOI

[11] M. Atiyah, Bull. Lond. Math. Soc., 14:1 (1982), 1–15 | DOI | MR | Zbl

[12] V. I. Man'ko, G. Marmo, E. C. G. Sudarshan, F. Zaccaria, Physica Scripta, 55:5 (1997), 528–541, arXiv: quant-ph/9612006 | DOI

[13] R. López-Peña, V. I. Man'ko, G. Marmo, Phys. Rev. A, 56:2 (1997), 1126–1130 | DOI

[14] N. N. Nekhoroshev, Regul. Chaotic Dyn., 7:3 (2002), 239–247 | DOI | MR | Zbl

[15] W. B. Gordon, J. Math. Mech., 19:2 (1969), 111–114 | MR | Zbl

[16] A. D'Avanzo, G. Marmo, Internat. J. Geom. Meth. Modern Phys., 2:1 (2005), 83–109 | DOI | MR | Zbl

[17] A. D'Avanzo, G. Marmo, A. Valentino, Internat. J. Geom. Meth. Modern Phys., 2:6 (2005), 1043–1062 | DOI | MR | Zbl

[18] A. Ibort, G. Marmo, “A new look at completely integrable systems and double Lie groups”, Secondary calculus and cohomological physics (Moscow, Russia, August 24–31, 1997), Contemporary Mathematics, 219, AMS, Providence RI, 1998, 159–172 | DOI | MR | Zbl