Electronic Journal of Differential Equations,
Vol. 1999(1999), No. 10, pp. 1-33.
Title: Implicit quasilinear differential systems: a geometrical approach
Authors: Miguel C. Munoz-Lecanda (Univ. Politecnica, Barcelona, Spain)
N. Roman-Roy (Univ. Politecnica, Barcelona, Spain)
Abstract:
This work is devoted to the study of systems of implicit quasilinear
differential equations. In general, no set of initial conditions is
admissible for the system. It is shown how to obtain a vector field
whose integral curves are the solution of the system, thus reducing
the system to one that is ordinary.
Using geometrical techniques, we give an algorithmic procedure in
order to solve these problems for systems of the form
$A({\bf x})\dot {\bf x} =\alpha ({\bf x})$ with $A({\bf x})$
being a singular matrix. As particular cases, we recover some results of
Hamiltonian and Lagrangian Mechanics. In addition, a detailed study of the
symmetries of these systems is carried out. This algorithm is applied to
several examples arising from technical applications related to control
theory.
Submitted November 30, 1998. Published April 1, 1999.
Math Subject Classification: 34C40, 57R25, 58A10, 58F99, 70Q05.
Key Words: Implicit differential equations; constrained systems;
vector fields; differentiable manifolds.