Involutive Division Technique: Some Generalizations and Optimizations

In this paper, in addition to the earlier introduced involutive divisions, we consider a new class of divisions induced by admissible monomial orderings. We prove that these divisions are noetherian and constructive. Thereby each of them allows one to compute an involutive Groebner basis of a polynomial ideal by sequentially examining multiplicative reductions of nonmultiplicative prolongations. We study dependence of involutive algorithms on the completion ordering. Based on properties of particular involutive divisions two computational optimizations are suggested. One of them consists in a special choice of the completion ordering. Another optimization is related to recomputing multiplicative and nonmultiplicative variables in the course of the algorithm.Comment: 19 page

The Cartan form for constrained Lagrangian systems and the nonholonomic Noether theorem

This paper deals with conservation laws for mechanical systems with nonholonomic constraints. It uses a Lagrangian formulation of nonholonomic systems and a Cartan form approach. We present what we believe to be the most general relations between symmetries and first integrals. We discuss the so-called nonholonomic Noether theorem in terms of our formalism, and we give applications to Riemannian submanifolds, to Lagrangians of mechanical type, and to the determination of quadratic first integrals.Comment: 25 page

Involution and Constrained Dynamics I: The Dirac Approach

We study the theory of systems with constraints from the point of view of the formal theory of partial differential equations. For finite-dimensional systems we show that the Dirac algorithm completes the equations of motion to an involutive system. We discuss the implications of this identification for field theories and argue that the involution analysis is more general and flexible than the Dirac approach. We also derive intrinsic expressions for the number of degrees of freedom.Comment: 28 pages, latex, no figure