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

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

Abstract

A necessary condition to be satisfied by n−1 vector fields in ℝ n 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 ℝ n .

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.

Keywords

Vector Field Smooth Vector Field Complete Trajectory Finite Term Linear Vector Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Authors and Affiliations

  1. 1.Département de Mathématiques et Informatique, UA CNRS 742, C. S. P.Université ParisNordVilletaneuseFrance
  2. 2.Départment de MathématiquesUniversité de RouenMont-Saint-Aignan CedexFrance

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