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1996 | 70 | 2 | 195-217
Tytuł artykułu

Liouvillian first integrals of homogeneouspolynomial 3-dimensional vector fields

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Given a 3-dimensional vector field V with coordinates $V_x$, $V_y$ and $V_z$ that are homogeneous polynomials in the ring k[x,y,z], we give a necessary and sufficient condition for the existence of a Liouvillian first integral of V which is homogeneous of degree 0. This condition is the existence of some 1-forms with coordinates in the ring k[x,y,z] enjoying precise properties; in particular, they have to be integrable in the sense of Pfaff and orthogonal to the vector field V. Thus, our theorem links the existence of an object that belongs to some level of an extension tower with the existence of objects defined by means of the base differential ring k[x,y,z]. A self-contained proof of this result is given in the language of differential algebra. This method of finding first integrals in a given class of functions is an extension of the compatibility method introduced by J.-M. Strelcyn and S. Wojciechowski; and an old method of Darboux is a special case of it. We discuss all these relations and argue for the practical interest of our characterization despite an old open algorithmic problem.
Słowa kluczowe
Rocznik
Tom
70
Numer
2
Strony
195-217
Opis fizyczny
Daty
wydano
1996
otrzymano
1995-07-10
Twórcy
  • GAGE, Centre de Mathématiques, Ecole Polytechnique, F-91128 Palaiseau Cedex, France
Bibliografia
  • [1] C. Camacho and A. Lins Neto, The topology of integrable differentiable forms near a singularity, Publ. Math. IHES 55 (1982), 5-36.
  • [2] H. Cartan, Formes différentielles, Hermann, Paris, 1967.
  • [3] D. Cerveau, Equations différentielles algébriques: remarques et problèmes, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), 665-680.
  • [4] D. Cerveau et F. Maghous, Feuilletages algébriques de $C^n$, C. R. Acad. Sci. Paris 303 (1986), 643-645.
  • [5] D. Cerveau et J. F. Mattei, Formes intégrables holomorphes singulières, Astérisque 97 (1982).
  • [6] G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré, Bull. Sci. Math. 2 (1878), 60-96, 123-144, 151-200.
  • [7] B. Grammaticos, J. Moulin Ollagnier, A. Ramani, J.-M. Strelcyn and S. Wojciechowski, Integrals of quadratic ordinary differential equations in $ℝ^3$: the Lotka-Volterra system, Phys. A 163 (1990), 683-722.
  • [8] J.-P. Jouanolou, Equations de Pfaff algébriques, Lecture Notes in Math. 708, Springer, Berlin, 1979.
  • [9] J. Moulin Ollagnier, A. Nowicki and J.-M. Strelcyn, On the non-existence of constants of derivations: the proof of a theorem of Jouanolou's and its development, Bull. Sci. Math. 119 (1995), 195-233.
  • [10] J. Moulin Ollagnier and J.-M. Strelcyn, On first integrals of linear systems, Frobenius integrability theorem and linear representations of Lie algebras, in: Bifurcations of Planar Vector Fields, Proceedings, Luminy 1989, J.-P. Françoise and R. Roussarie (eds.), Lecture Notes in Math. 1455, Springer, Berlin, 1991.
  • [11] H. Poincaré, Sur l'intégration algébrique des équations différentielles, C. R. Acad. Sci. Paris 112 (1891), 761-764; reprinted in Œ uvres, tome III, Gauthier-Villars, Paris, 1965, 32-34.
  • [12] H. Poincaré, Sur l'intégration algébrique des équations différentielles du premier ordre et du premier degré, Rend. Circ. Mat. Palermo 5 (1891), 161-191; reprinted in Œ uvres, tome III, Gauthier-Villars, Paris, 1965, 35-58.
  • [13] H. Poincaré, Sur l'intégration algébrique des équations différentielles du premier ordre et du premier degré, Rend. Circ. Mat. Palermo 11 (1897), 193-239; reprinted in Œ uvres, tome III, Gauthier-Villars, Paris, 1965, 59-94.
  • [14] M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc. 279 (1983), 215-229.
  • [15] R. H. Risch, The problem of integration in finite terms, Trans. Amer. Math. Soc. 139 (1969), 167-189.
  • [16] M. Rosenlicht, On Liouville's theory of elementary functions, Pacific J. Math. 65 (1976), 485-492.
  • [17] M. F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333 (1992), 673-688.
  • [18] J.-M. Strelcyn and S. Wojciechowski, A method of finding integrals of 3-dimensional dynamical systems, Phys. Lett. A 133 (1988), 207-212.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-cmv70i2p195bwm
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