DIFFERENTIAL EQUATIONS

The geometric approach to the study of differential equations goes back to Sophus Lie and Elie Cartan. According to the modern inter-pretation of this approach, based on the notion ofjet space, we consider a differential equation as a submanifold in the jet space with induced geometric structure. Using the methods of filtered manifolds developed in works of Tana-ka $[3, 4] $ and Morimoto [2], we construct a characteristic Cartan con-nection, naturally associated with any system of $m $ equations of the $(n+1)$-th order whenever $m\geq 2, $ $n\geq 1 $ or $m=1, $ $n\geq 2 $. Then we compute the compete set of fundamental invariants which appear as coefficients of the curvature tensor. Here by fundamental invariants of ordinary differential equations we understand relative invariants with respect to the contact transformations which generate the set of all invariants of a given ODE. Note that in the case of single second order ODE there is a classical result of Sophus Lie showing that all ODE’s of the second order are contact equivalent and have an infinite dimensional symmetry algebra, which makes it impossible to construct a characteristic connection in this particular case. Theorem 1 ([1]). With any system of $m $ ordinary differential equations of the $(n+1) $-th order, where $m\geq 2, $ $n\geq 1 $ or $m=1, $ $n\geq 2 $ , there is naturally associated a Cartan connection with model $G/H $ , where for $m=1, $ $n=2 $: $G=SP(4, \mathbb{R}), $ $H $ is the Borel subgroup of $G $