Straighten out coordinates for volume-preserving actions

  • Mário BessaEmail author
  • Pedro Morais


In this short note we obtain a canonical form for commuting divergence-free vector fields.


Divergence-free \({\mathbb {R}}^n\) actions Foliations Flowbox coordinates 

Mathematics Subject Classification

Primary 37C10 34A26 Secondary 57R30 34C20 



The authors would like to thank the anonymous referee for the careful reading of the manuscript and for giving helpful comments and suggestions. The authors were partially supported by FCT - ‘Fundação para a Ciência e a Tecnologia’, through Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, project UID/MAT/00212/2019. MB was partially supported by the Project ‘New trends in Lyapunov exponents’ (PTDC/MAT-PUR/29126/2017). MB would also like to thank CMUP for providing the necessary conditions in which this work was developed.


  1. 1.
    Abraham, R., Marsden, J.E.: Foundations of Mechanics. Benjamin/Cummings Publishing Co., Inc, Advanced Book Program, Reading, Mass. (1978)zbMATHGoogle Scholar
  2. 2.
    Barbarosie, C.: Representation of divergence-free vector fields. Quart. Appl. Math. 69(2), 309–316 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bessa, M.: The Lyapunov exponents of zero divergence three-dimensional vector fields. Ergod. Theory Dyn. Syst. 27(5), 1445–1472 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bessa, M., Rocha, J.: Removing zero Lyapunov exponents in volume-preserving flows. Nonlinearity 20(4), 1007–1016 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bessa, M., Lopes-Dias, J.: Generic dynamics of 4-dimensional \(C^2\) Hamiltonian systems. Commun. Math. Phys. 281(3), 597–619 (2008)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bryant, R.L.: An introduction to Lie groups and symplectic geometry. In: Freed, D.S., Uhlembeck, K.K. (eds.) Geometry and Quantum Field Theory, pp. 5–181. Amer. Math. Soc., Providence, RI (1995)Google Scholar
  7. 7.
    Cabral, H.E.: On the Hamiltonian flow box theorem. Qual. Theory Dyn. Syst. 12(1), 5–9 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Castro, F., Oliveira, F.: On the transitivity of invariant manifolds of conservative flows, Preprint (2015). arXiv:1503.00182
  9. 9.
    Dacorogna, B., Moser, J.: On a partial differential equation involving the Jacobian determinant. Ann. Inst. Henri Poincaré 7(1), 1–26 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lee, J.M.: Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Springer, New York (2003)CrossRefGoogle Scholar
  11. 11.
    Meyer, K.R., Hall, G.R., Offin, D.: Introduction to Hamiltonian Dynamical Systems and the \(N\)-Body Problem, 2nd edn. Springer, New York (2009)CrossRefzbMATHGoogle Scholar
  12. 12.
    Robinson, C.: Lectures on Hamiltonian Systems. Monograf. Mat, IMPA, London (1971)Google Scholar

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Authors and Affiliations

  1. 1.CMA-UBI, Departamento de Matemática da Universidade da Beira InteriorCovilhãPortugal

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