Geodesic
Geometry of third order ODE systems
Archivum mathematicum, Tome 46 (2010) no. 5, pp. 351-361 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 17B56, 34A26, 53B15
Keywords: geometry of ordinary differential equations; normal Cartan connections, cohomology of Lie algebras 
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     author = {Medvedev, Alexandr},
     title = {Geometry of third order {ODE} systems},
     journal = {Archivum mathematicum},
     pages = {351--361},
     year = {2010},
     volume = {46},
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     mrnumber = {2753989},
     zbl = {1249.34024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2010_46_5_a5/}
}
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Medvedev, Alexandr. Geometry of third order ODE systems. Archivum mathematicum, Tome 46 (2010) no. 5, pp. 351-361. http://geodesic.mathdoc.fr/item/ARM_2010_46_5_a5/

[1] Doubrov, B.: Contact trivialization of ordinary differential equations. Proc. Conf., Opava (Czech Republic), Differential Geometry and Its Applications, 2001, pp. 73–84. | MR | Zbl

[2] Doubrov, B.: Generalized Wilczynski invariants for non-linear ordinary differential equations. Symmetries and overdetermined systems of partial differential equations. IMA Vol. Math. Appl. 144 (2008), 25–40. | DOI | MR

[3] Doubrov, B., Komrakov, B., Morimoto, T.: Equivalence of holonomic differential equations. Lobachevskij J. Math. 3 (1999), 39–71. | MR | Zbl

[4] Fuks, D. B.: Cohomology of infinite-dimensional Lie algebras. Contemporary Soviet Mathematics, New York: Consultants Bureau, 1986. | MR | Zbl

[5] Kobayashi, S., Nagano, T.: On filtered Lie algebras and geometric structures III. J. Math. Mech. 14 (1965), 679–706. | MR

[6] Kostant, B.: Lie algebra cohomology and the generalized Borel–Weil theorem. Ann. of Math. (2) 24 (1961), 329–387. | DOI | MR | Zbl

[7] Morimoto, T.: Geometric structures on filtered manifolds. Hokkaido Math. J. 22 (1993), 263–347. | MR | Zbl

[8] Tanaka, N.: On differential systems, graded Lie algebras and pseudo-groups. J. Math. Kyoto Univ. 10 (1970), 1–82. | MR | Zbl

[9] Tanaka, N.: On the equivalence problems associated with simple graded Lie algebras. Hokkaido Math. J. 8 (1979), 23–84. | MR | Zbl

[10] Tanaka, N.: Geometric theory of ordinary differential equations. Report of Grant-in-Aid for Scientific Research MESC Japan (1989).

[11] Yamaguchi, K.: Differential systems associated with simple graded Lie algebras. Adv. Stud. Pure Math. 22 (1993), 413–494. | MR | Zbl