Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 035, 9 pages      arXiv:1206.3005      http://dx.doi.org/10.3842/SIGMA.2012.035

A Note on the First Integrals of Vector Fields with Integrating Factors and Normalizers

Jaume Llibre a and Daniel Peralta-Salas b
a) Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
b) Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera 13-15, 28049 Madrid, Spain

Received February 16, 2012, in final form June 12, 2012; Published online June 14, 2012

Abstract
We prove a sufficient condition for the existence of explicit first integrals for vector fields which admit an integrating factor. This theorem recovers and extends previous results in the literature on the integrability of vector fields which are volume preserving and possess nontrivial normalizers. Our approach is geometric and coordinate-free and hence it works on any smooth orientable manifold.

Key words: first integral; vector field; integrating factor; normalizer.

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References

  1. Abraham R., Marsden J.E., Foundations of mechanics, 2nd ed., Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, Mass., 1978.
  2. Arnold V.I., Kozlov V.V., Neishtadt A.I., Mathematical aspects of classical and celestial mechanics, Springer-Verlag, Berlin, 1997.
  3. Bluman G.W., Anco S.C., Symmetry and integration methods for differential equations, Applied Mathematical Sciences, Vol. 154, Springer-Verlag, New York, 2002.
  4. Campoamor-Stursberg O.R., González-Gascón F., Peralta-Salas D., Dynamical systems embedded into Lie algebras, J. Math. Phys. 42 (2001), 5741-5752.
  5. García I.A., Grau M., A survey on the inverse integrating factor, Qual. Theory Dyn. Syst. 9 (2010), 115-166, arXiv:0903.0941.
  6. González-Gascón F., On a new first integral of certain dynamical systems, Phys. Lett. A 61 (1977), 375-376.
  7. González-Gascón F., Peralta-Salas D., Symmetries and first integrals of divergence-free R3 vector fields, Internat. J. Non-Linear Mech. 35 (2000), 589-596.
  8. Goriely A., Integrability and nonintegrability of dynamical systems, Advanced Series in Nonlinear Dynamics, Vol. 19, World Scientific Publishing Co. Inc., River Edge, NJ, 2001.
  9. Haller G., Mezic I., Reduction of three-dimensional, volume-preserving flows with symmetry, Nonlinearity 11 (1998), 319-339.
  10. Hojman S.A., A new conservation law constructed without using either Lagrangians or Hamiltonians, J. Phys. A: Math. Gen. 25 (1992), L291-L295.
  11. Huang D., A coordinate-free reduction for flows on the volume manifold, Appl. Math. Lett. 17 (2004), 17-22.
  12. Kozlov V.V., Remarks on a Lie theorem on the integrability of differential equations in closed form, Differ. Equ. 41 (2005), 588-590.
  13. Marcelli M., Nucci M.C., Lie point symmetries and first integrals: the Kowalevski top, J. Math. Phys. 44 (2003), 2111-2132, nlin.SI/0201023.
  14. Merker J., Theory of transformation groups, by S. Lie and F. Engel (Vol. I, 1888). Modern presentation and english translation, arXiv:1003.3202.
  15. Mezic I., Wiggins S., On the integrability and perturbation of three-dimensional fluid flows with symmetry, J. Nonlinear Sci. 4 (1994), 157-194.
  16. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, 2nd ed., Springer-Verlag, New York, 1993.
  17. Peralta-Salas D., Period function and normalizers of vector fields in Rn with n−1 first integrals, J. Differential Equations 244 (2008), 1287-1303.
  18. Prelle M.J., Singer M.F., Elementary first integrals of differential equations, Trans. Amer. Math. Soc. 279 (1983), 215-229.
  19. Prince G., Comment on "Period function and normalizers of vector fields in Rn with n−1 first integrals", J. Differential Equations 246 (2009), 3750-3753.
  20. Sherring J., Prince G., Geometric aspects of reduction of order, Trans. Amer. Math. Soc. 334 (1992), 433-453.
  21. Walcher S., Plane polynomial vector fields with prescribed invariant curves, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), 633-649.

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