Geometry of third order ODE systems

Alexandr Medvedev

Archivum Mathematicum (2010)

  • Volume: 046, Issue: 5, page 351-361
  • ISSN: 0044-8753

Abstract

top
We compute cohomology spaces of Lie algebras that describe differential invariants of third order ordinary differential equations. We prove that the algebra of all differential invariants is generated by 2 tensorial invariants of order 2, one invariant of order 3 and one invariant of order 4. The main computational tool is a Serre-Hochschild spectral sequence and the representation theory of semisimple Lie algebras. We compute differential invariants up to degree 2 as application.

How to cite

top

Medvedev, Alexandr. "Geometry of third order ODE systems." Archivum Mathematicum 046.5 (2010): 351-361. <http://eudml.org/doc/116498>.

@article{Medvedev2010,
abstract = {We compute cohomology spaces of Lie algebras that describe differential invariants of third order ordinary differential equations. We prove that the algebra of all differential invariants is generated by 2 tensorial invariants of order 2, one invariant of order 3 and one invariant of order 4. The main computational tool is a Serre-Hochschild spectral sequence and the representation theory of semisimple Lie algebras. We compute differential invariants up to degree 2 as application.},
author = {Medvedev, Alexandr},
journal = {Archivum Mathematicum},
keywords = {geometry of ordinary differential equations; normal Cartan connections; cohomology of Lie algebras; geometry of ordinary differential equations; normal Cartan connections; cohomology of Lie algebras},
language = {eng},
number = {5},
pages = {351-361},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Geometry of third order ODE systems},
url = {http://eudml.org/doc/116498},
volume = {046},
year = {2010},
}

TY - JOUR
AU - Medvedev, Alexandr
TI - Geometry of third order ODE systems
JO - Archivum Mathematicum
PY - 2010
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 046
IS - 5
SP - 351
EP - 361
AB - We compute cohomology spaces of Lie algebras that describe differential invariants of third order ordinary differential equations. We prove that the algebra of all differential invariants is generated by 2 tensorial invariants of order 2, one invariant of order 3 and one invariant of order 4. The main computational tool is a Serre-Hochschild spectral sequence and the representation theory of semisimple Lie algebras. We compute differential invariants up to degree 2 as application.
LA - eng
KW - geometry of ordinary differential equations; normal Cartan connections; cohomology of Lie algebras; geometry of ordinary differential equations; normal Cartan connections; cohomology of Lie algebras
UR - http://eudml.org/doc/116498
ER -

References

top
  1. Doubrov, B., Contact trivialization of ordinary differential equations, Proc. Conf., Opava (Czech Republic), Differential Geometry and Its Applications, 2001, pp. 73–84. (2001) Zbl1047.34040MR1978764
  2. Doubrov, B., 10.1007/978-0-387-73831-4_2, IMA Vol. Math. Appl. 144 (2008), 25–40. (2008) MR2384704DOI10.1007/978-0-387-73831-4_2
  3. Doubrov, B., Komrakov, B., Morimoto, T., Equivalence of holonomic differential equations, Lobachevskij J. Math. 3 (1999), 39–71. (1999) Zbl0937.37051MR1743131
  4. Fuks, D. B., Cohomology of infinite-dimensional Lie algebras, Contemporary Soviet Mathematics, New York: Consultants Bureau, 1986. (1986) Zbl0667.17005MR0874337
  5. Kobayashi, S., Nagano, T., On filtered Lie algebras and geometric structures III, J. Math. Mech. 14 (1965), 679–706. (1965) MR0188364
  6. Kostant, B., 10.2307/1970237, Ann. of Math. (2) 24 (1961), 329–387. (1961) Zbl0134.03501MR0142696DOI10.2307/1970237
  7. Morimoto, T., Geometric structures on filtered manifolds, Hokkaido Math. J. 22 (1993), 263–347. (1993) Zbl0801.53019MR1245130
  8. Tanaka, N., On differential systems, graded Lie algebras and pseudo-groups, J. Math. Kyoto Univ. 10 (1970), 1–82. (1970) Zbl0206.50503MR0266258
  9. Tanaka, N., On the equivalence problems associated with simple graded Lie algebras, Hokkaido Math. J. 8 (1979), 23–84. (1979) Zbl0409.17013MR0533089
  10. Tanaka, N., Geometric theory of ordinary differential equations, Report of Grant-in-Aid for Scientific Research MESC Japan (1989). (1989) 
  11. Yamaguchi, K., Differential systems associated with simple graded Lie algebras, Adv. Stud. Pure Math. 22 (1993), 413–494. (1993) Zbl0812.17018MR1274961

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.