Computations involving differential operators and their actions on functions

The algorithms derived by Grossmann and Larson (1989) are further developed for rewriting expressions involving differential operators. The differential operators involved arise in the local analysis of nonlinear dynamical systems. These algorithms are extended in two different directions: the algorithms are generalized so that they apply to differential operators on groups and the data structures and algorithms are developed to compute symbolically the action of differential operators on functions. Both of these generalizations are needed for applications

An almost Poisson structure for the generalized rigid body equations

In this paper we introduce almost Poisson structures on Lie groups which generalize Poisson structures based on the use of the classical Yang-Baxter identity. Almost Poisson structures fail to be Poisson structures in the sense that they do not satisfy the Jacobi identity.In the case of cross products of Lie groups, we show that an almost Poisson structure can be used to derive a system which is intimately related to a fundamental Hamiltonian integrable system — the generalized rigid body equations

Discrete rigid body dynamics and optimal control

We analyze an alternative formulation of the rigid body equations, their relationship with the discrete rigid body equations of Moser-Veselov (1991) and their formulation as an optimal control problem. In addition we discuss a general class of discrete optimal control problems

Adjoint and Hamiltonian input-output differential equations

Based on developments in the theory of variational and Hamiltonian control systems by Crouch and van der Schaft (1987), this paper answers two questions: given an input-output differential equation description of a nonlinear system, what is the adjoint variational system in input-output differential form and what are the conditions for the system to be Hamiltonian, i.e., such that the variational and the adjoint variational systems coincide? This resulting set of conditions is then used to generalize classical conditions such as the well-known Helmholtz conditions for the inverse problem in classical mechanics

A variational problem on Stiefel manifolds

In their paper on discrete analogues of some classical systems such as the rigid body and the geodesic flow on an ellipsoid, Moser and Veselov introduced their analysis in the general context of flows on Stiefel manifolds. We consider here a general class of continuous time, quadratic cost, optimal control problems on Stiefel manifolds, which in the extreme dimensions again yield these classical physical geodesic flows. We have already shown that this optimal control setting gives a new symmetric representation of the rigid body flow and in this paper we extend this representation to the geodesic flow on the ellipsoid and the more general Stiefel manifold case. The metric we choose on the Stiefel manifolds is the same as that used in the symmetric representation of the rigid body flow and that used by Moser and Veselov. In the extreme cases of the ellipsoid and the rigid body, the geodesic flows are known to be integrable. We obtain the extremal flows using both variational and optimal control approaches and elucidate the structure of the flows on general Stiefel manifolds.Comment: 30 page