The Integrability of Lie-invariant Geometric Objects Generated by Ideals in the Grassmann Algebra

Abstract

We investigate closed ideals in the Grassmann algebra serving as bases of Lie-invariant geometric objects studied before by E.Cartan. Especially, the E.Cartan theory is enlarged for Lax integrable nonlinear dynamical systems to be treated in the frame work of the Wahlquist Estabrook prolongation structures on jet-manifolds and Cartan-Ehresmann connection theory on fibered spaces. General structure of integrable one-forms augmenting the two-forms associated with a closed ideal in the Grassmann algebra is studied in great detail. An effective Maurer-Cartan one-forms construction is suggested that is very useful for applications. As an example of application the developed Lie-invariant geometric object theory for the Burgers nonlinear dynamical system is considered having given rise to finding an explicit form of the associated Lax type representation. 1 General setting It is well known [1, 4] that motion planning, numerically controlled machining and robotics are just a few of many areas of manufacturing automation in which the analysis and representation of swept volumes plays a crucial role. The swept volume modeling is also an important part of task-oriented robot motion planning. A typical motion planning problem consists in a collection of objects moving around obstacles from an initial to a final configuration. This may include in particular, solving the collision detecting problem. When a solid object undergoes a rigid motion, the totality of points through which it passed constitutes a region in space called the swept volume. To describe the geometrical structure of the swept volume we pose this problem as one of geometric study of some manifold swept by surface points using powerful tools from both modern differential ge-ometry and nonlinear dynamical systems theory [2-4, 7, 8] on manifolds. For some special Copyright c©1998 by the Authors