Operator of invariant differentiation and its application for integrating systems of ordinary differential equations

Abstract

We propose an algorithm for integrating $n$-th order ordinary differential equations (ODE) admitting $n$-dimensional Lie algebras of operators. The algorithm is based on invariant representation of the equations by the invariants of the admitted Lie algebra and application of an operator of invariant differentiation of special type. We show that in the case of scalar equations this method is equivalent to the known order reduction methods. We study an applicability of the suggested algorithm to the systems of $m$ $k$-th order ODEs admitting $km$-dimensional Lie algebras of operators. For the admitted Lie algebra we obtain a condition ensuring the possibility to construct the operator of invariant differentiation of a special type and to reduce the order of the considered system of ODEs. This condition is the implication of the existence of nontrivial solutions to the systems of linear algebraic equations, where the coefficients are the structural constants of the Lie algebra. We present an algorithm for constructing the $(km-1)$-dimensional Lie algebra for the reduced system. The suggested approach is applied for integrating the systems of two second order equations.