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


SIGMA 3 (2007), 103, 7 pages      arXiv:0711.0814      https://doi.org/10.3842/SIGMA.2007.103
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Geometric Linearization of Ordinary Differential Equations

Asghar Qadir
Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, Campus of College of Electrical and Mechanical Engineering, Peshawar Road, Rawalpindi, Pakistan

Received August 13, 2007, in final form October 19, 2007; Published online November 06, 2007; References [17−21] updated November 11, 2007

Abstract
The linearizability of differential equations was first considered by Lie for scalar second order semi-linear ordinary differential equations. Since then there has been considerable work done on the algebraic classification of linearizable equations and even on systems of equations. However, little has been done in the way of providing explicit criteria to determine their linearizability. Using the connection between isometries and symmetries of the system of geodesic equations criteria were established for second order quadratically and cubically semi-linear equations and for systems of equations. The connection was proved for maximally symmetric spaces and a conjecture was put forward for other cases. Here the criteria are briefly reviewed and the conjecture is proved.

Key words: differential equations; geodesics; geometry; linearizability; linearization.

pdf (185 kb)   ps (139 kb)   tex (12 kb)

References

  1. Lie S., Theorie der transformationsgruppen, Math. Ann. 16 (1880), 441-528.
  2. Lie S., Klassification und Integration von gewöhnlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten, Arch. Math. Naturv. 9 (1883), 371-393.
  3. Olver P.J., Applications of Lie groups to differential equations, Springer, New York, 1986.
  4. Wafo Soh C., Mahomed F.M., Symmetry breaking for a system of two linear second-order ordinary differential equations, Nonlinear Dynamics 22 (2000), 121-133.
  5. Mahomed F.M., Leach P.G.L., Symmetry Lie algebras of nth order ordinary differential equations, J. Math. Anal. Appl. 151 (1990), 80-107.
  6. Chern S.S., Sur la geometrie d'une equation differentielle du troiseme orde, C.R. Acad. Sci. Paris 204 (1937), 1227-1229.
  7. Chern S.S., The geometry of the differential equation y¢¢¢=F(x,y,y¢,y¢¢), Sci. Rep. Nat. Tsing Hua Univ. 4 (1940), 97-111.
  8. Grebot G., The linearization of third order ODEs, Preprint, 1996.
  9. Grebot G., The characterization of third order ordinary differential equations admitting a transitive fibre-preserving point symmetry group, J. Math. Anal. Appl. 206 (1997), 364-388.
  10. Neut S., Petitot M., La géométrie de l'équation y¢¢¢=f(x,y,y¢,y¢¢), C.R. Acad. Sci. Paris Sér I 335 (2002), 515-518.
  11. Ibragimov N.H., Meleshko S.V., Linearization of third-order ordinary differential equations by point and contact transformations, J. Math. Anal. Appl. 308 (2005), 266-289.
  12. Meleshko S.V., On linearization of third-order ordinary differential equations, J. Phys. A: Math. Gen. Math. 39 (2006), 15135-15145.
  13. Aminova A.V., Aminov N.A.-M., Projective geometry of systems of differential equations: general conceptions, Tensor NS 62 (2000), 65-85.
  14. Bryant R.L., Manno G., Matveev V.S., A solution of a problem of Sophus Lie: normal forms of 2-dim metrics admitting two projective vector fields, arXiv:0705.3592.
  15. Feroze T., Mahomed F.M., Qadir A., The connection between isometries and symmetries of geodesic equations of the underlying spaces, Nonlinear Dynamics 45 (2006), 65-74.
  16. Mahomed F.M., Qadir A., Linearization criteria for a system of second-order quadratically semi-linear ordinary differential equations, Nonlinear Dynamics 48 (2007), 417-422.
  17. Mahomed F.M., Qadir A., Invariant linearization criteria for systems of cubically semi-linear second-order ordinary differential equations, arXiv:0711.1213.
  18. Fredericks E., Mahomed F.M., Momoniat E., Qadir A., Constructing a space from the system of geodesic equations, arXiv:0711.1217.
  19. Mahomed F.M., Qadir A., Linearizability criteria for a class of third order semi-linear ODEs, arXiv:0711.1214.
  20. Mahomed F.M., Qadir A., Conditional linearizability criteria for scalar fourth order semi-linear ordinary differential equations, arXiv:0711.1222.
  21. Mahomed F.M., Naeem I., Qadir A., Conditional linearizability criteria for a system of third-order ordinary differential equations, arXiv:0711.1215.

Previous article   Next article   Contents of Volume 3 (2007)