Solutions of singular constrained differential equations: A generalization of circuits containing capacitor-only loops and inductor-only cutsets

Abstract

Employing the theory of differentiable manifolds we give a geometric coordinate-free description of constrained differential equations (CDE), which are usually thought of as systems of simultaneous differential and algebraic equations. We regard the algebraic constraints as defining a differentiable manifold, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Sigma</tex> , and regard solutions of the CDE as curves in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Sigma</tex> . Our main contribution in this paper is to characterize geometrically a particular class of singular constrained differential equations for which consistency forces solutions to lie in a proper subset of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Sigma</tex> . Constrained differential equations describing electric circuits having capacitor-only loops and/or inductor-only cutsets are shown to be of the above type.