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On first integrals of linear systems, Frobenius integrability theorem and linear representations of Lie algebras

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Bifurcations of Planar Vector Fields

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1455))

Abstract

A necessary condition to be satisfied by n−1 vector fields in ℝn in order to have a common first integral is supplied by the compatibility condition of Frobenius integrability theorem. This condition is also generically sufficient for the local existence of such a common first integral. We study here the question of the existence of a global common first integral for compatible linear vector fields in ℝn.

For the dimension 3, we prove that any two compatible linear vector fields have a common global first integral.

On the contrary, we give an example for the dimension 4, in which three compatible linear vector fields cannot have a common global first integral.

This leads us to ask many simple and natural questions, some of them about representations of Lie algebras by Lie algebras of linear vector fields.

Some historical comments and abundant references are also provided.

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Authors and Affiliations

  1. Département de Mathématiques et Informatique, UA CNRS 742, C. S. P., Université ParisNord, Avenue J. B. Clément, 93430, Villetaneuse, France

    Jean Moulin Ollagnier

  2. Départment de Mathématiques, Université de Rouen, UA CNRS 742 & 1378, B.P. 118, 76134, Mont-Saint-Aignan Cedex, France

    Jean-Marie Strelcyn

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  1. Jean Moulin Ollagnier
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  2. Jean-Marie Strelcyn
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Jean-Pierre Françoise Robert Roussarie

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Ollagnier, J.M., Strelcyn, JM. (1990). On first integrals of linear systems, Frobenius integrability theorem and linear representations of Lie algebras. In: Françoise, JP., Roussarie, R. (eds) Bifurcations of Planar Vector Fields. Lecture Notes in Mathematics, vol 1455. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085396

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  • DOI: https://doi.org/10.1007/BFb0085396

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