Linearization criteria for a system of second-order ordinary differential equations

Abstract

Firstly, we prove two linearization criteria for a system of two second-order ordinary differential equations (odes). The first states that a system of two non-linear second-order odes is reducible via a point transformation to a linear system if and only if it admits the four-dimensional abelian Lie algebra L 4,1 : [X i,X j]=0, i, j=1,…, 4 . The second states that in order for a system of two non-linear second-order odes to be linearizable, it is necessary and sufficient that it admits the four-dimensional Lie algebra L 4,2 : [X i,X j]=0, [X i,X 4]=X i, i, j=1, 2, 3 . The approach used is constructive and enables one to explicitly work out the transformation that leads to linearization. Secondly, we give conditions under which a system of two second-order non-linear odes is reducible to the free particle equations x″=0, y″=0 . These linearization criteria are then generalized to a system of n( n>2) second-order odes. Finally, we give examples of how one can effect linearization for a system.