# Geometry of third order ODE systems

## Alexandr Medvedev

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

E-mail: sasha.medvedev@gmail.com

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.

AMSclassification: primary 34A26; secondary 17B56, 53B15.

Keywords: geometry of ordinary differential equations, normal Cartan connections, cohomology of Lie algebras.