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Summary. This paper is concerned with the problem of developing numerical integration algorithms for differential equations that, when viewed as equations in some Euclidean space, naturally evolve on some embedded submanifold. It is desired to construct algorithms whose iterates also evolve on the same manifold. These algorithms can therefore be viewed as integrating ordinary differential equations on manifolds. The basic method "decouples" the computation of flows on the submanifold from the numerical integration process. It is shown that two classes of single-step and multistep algorithms can be posed and analyzed theoretically, using the concept of "freezing" the coefficients of differential operators obtained from the defining vector field. Explicit third-order algorithms are derived, with additional equations augmenting those of their classical counterparts, obtained from "obstructions" defined by nonvanishing Lie brackets.

ABSTRACT. We consider the dynamic interpolation problem for nonlinear control systems modeled by second-order differential equations whose configuration space is a Riem~nnlan manifold M. In this problem we are given an ordered set of points in M and would like to generate a trajectory of the system through the application of suitable control functions, so that the resulting trajectory in configuration space interpolates the given set of points. We also impose smoothness constraints on the trajectory and typically ask that the trajectory be also optimal with respect to some physically interesting cost function. Here we are interested in the situation where the trajectory is twice continuously differentlable and the Lagrangian in the optimization problem is given by the norm squared acceleration along the trajectory. The special cases where M is a connected and compact Lie group or a homogeneous symmetric space are studied in more detail.

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This paper presents the findings of an Engineering Curriculum Task Force of the College of Engineering and Applied Sciences at Arizona State University. The Task Force's charge was to explore changes that would better prepare baccalaureate‐level engineers for the practice of their profession in the next decade. The generic data developed in the process used by the Task Force are given here. For example, a set of ten important attributes deemed desirable for newly graduated engineers and produced by education, is presented. Also, the rankings by industry, alumni, students, and faculty, of the relative importance of each of these ten attributes and the performance of new graduates in each, are given. Generic curriculum features necessary for successful generation of these attributes are discussed. Finally, the results of an alumni survey are presented which show the overwhelming support for a broad‐based undergraduate program by graduates of all degree programs.

This paper analyses continuous and discrete versions of the generalized rigid body equations and the role of these equations in numerical analysis, optimal control and integrable Hamiltonian systems. In particular, we present a symmetric representation of the rigid body equations on the Cartesian product SO(n) × SO(n) and study its associated symplectic structure. We describe the relationship of these ideas with the Moser-Veselov theory of discrete integrable systems and with the theory of variational symplectic integrators. Preliminary work on the ideas discussed in this paper may be found in Bloch et al (Bloch A M, Crouch P, Marsden J E and Ratiu T S 1998 Proc. IEEE Conf. on Decision and Control 37 2249-54).

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