Index and Decomposition of Nonlinear Implicit Differential Equations

Abstract

In engineering, as well as in physics, one often encounters implicit systems of ordinary differential equations of the form Fi(ẏ1…ẏm, y1…ym) = 0, i = 1,…, m, in the unknowns y1…ym, where the Jacobian matrix (∂Fi/∂ẏj)i,j is identically singular. We first state a condition under which the above equation may be solved. Then, we give a decomposition, which provides as a byproduct a clear-cut definition of the index. Since the Fi's must be differentiated an arbitrary number of times, we employ tools stemming from the differential geometry of infinite jets and prolongations like Lie-Bäcklund morphisms.