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


SIGMA 7 (2011), 076, 15 pages      arXiv:1104.0965      http://dx.doi.org/10.3842/SIGMA.2011.076

Third Order ODEs Systems and Its Characteristic Connections

Alexandr Medvedev
Faculty of Applied Mathematics, Belarusian State University, 4, Nezavisimosti Ave., 220030, Minsk, Republic of Belarus

Received April 20, 2011, in final form July 27, 2011; Published online August 03, 2011

Abstract
We compute the characteristic Cartan connection associated with a system of third order ODEs. Our connection is different from Tanaka normal one, but still is uniquely associated with the system of third order ODEs. This allows us to find all fundamental invariants of a system of third order ODEs and, in particular, determine when a system of third order ODEs is trivializable. As application differential invariants of equations on circles in Rn are computed.

Key words: geometry of ordinary differential equations; normal Cartan connections.

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References

  1. Chern S.-S., The geometry of the differential equation y'''=F(x,y,y',y''), Sci. Rep. Nat. Tsing Hua Univ. (A) 4 (1940), 97-111.
  2. Doubrov B., Contact trivialization of ordinary differential equations, in Differential Geometry and Its Applications (Opava, 2001), Math. Publ., Vol. 3, Silesian Univ. Opava, Opava, 2001, 73-84.
  3. Doubrov B., Komrakov B., Morimoto T., Equivalence of holonomic differential equations, Lobachevskii J. Math. 3 (1999), 39-71.
  4. Fels M., The equivalence problem for systems of second-order ordinary differential equations, Proc. London Math. Soc. 71 (1995), 221-240.
  5. Lie S., Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen, Leipzig, Teubner, 1891.
  6. Medvedev A., Geometry of third order ODE systems, Arch. Math. (Brno) 46 (2010), 351-361.
  7. Morimoto T., Geometric structures on filtered manifolds, Hokkaido Math. J. 22 (1993), 263-347.
  8. Sato H., Yoshikawa A.Y., Third order ordinary differential equations and Legendre connections, J. Math. Soc. Japan 50 (1998), 993-1013.
  9. Tanaka N., On differential systems, graded Lie algebras and pseudo-groups, J. Math. Kyoto. Univ. 10 (1970), 1-82.
  10. Tanaka N., On the equivalence problems associated with simple graded Lie algebras, Hokkaido Math. J. 8 (1979), 23-84.
  11. Tresse M.A., Détermination des invariants ponctuels de l'équation différentielle ordinaire du second ordre y'' = ω(x,y,y'), Leipzig, Hirzel, 1896.
  12. Yano K., The theory of Lie derivatives and its applications, North Holland Publishing Co., Amsterdam, 1957.

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