The theory and applications of generalized connections and moving frames for nonfree actions is studied, with particular emphasis on geometric integration methods for the numerical integration of differential equations with symmetries and nonlinear constraints. Classical connections are defined only for free actions; the normalization condition would force the Lie algebra-valued connection form to be singular at singular points of the group action. However, if the connection form is regarded simply as a mechanism for specifying an equivariant projection onto the tangent space to the group orbit, the connection form can be replaced with a smooth one form taking values in the dual of the Lie algebra. Several key results from the classical theory of connections on principal bundles are generalized to partial connections. In support of this theoretical work, and as an application of it, a variety of mechanical systems are studied, including models from molecular dynamics, celestial mechanics, and micromagnetism. The primary application is the design and analysis of geometric integration schemes, that is, algorithms for the numerical approximation of trajectories of differential equations that capture some essential features of the exact solutions, independent of the overall accuracy of the approximations. The decomposition of tangent vectors into vertical (rigid) and horizontal (internal) components given by a partial connection form can be used to construct geometric integrators combining Lie group methods with traditional numerical integration schemes. Such decompositions also facilitate the development of geometrically meaningful performance criteria for competing numerical schemes.
A `connection' is, roughly speaking, a consistent way of describing the evolution of a mechanical system as a combination of `rigid' motions determined by the symmetry group and `shape-changing' motions. (In fact, connections can be defined for more general problems using appropriate mathematical analogs of rigid motions and shape.) Such decompositions have long played a central role in mechanics, geometry, and mathematical physics. For example, connections can be used in engineering applications to efficiently predict the orientation of a flexible system (e.g. a complex space structure), when a detailed simulation of the motion of the system is intractable. Symmetric states (e.g. axisymmetric shapes) are of tremendous practical and theoretical importance; almost all man-made objects have some symmetry and many are highly symmetric. However, connections for systems with symmetric configurations have been introduced only recently, by the investigator and coworkers, and many crucial features remain to be explored. A diverse assortment of applications, relevant to computer design and manufacture, telecommunications, and medicine, motivate and guide the new theory. For example, generalized connections can be used in the design of efficient computer simulations of multibody systems, which are used in satellite design and control, biomechanics, and robotics. The development of geometric algorithms that can be implemented as `after-market' modifications of conventional algorithms facilitates the adoption of innovative methods in industrial and applied research settings by minimizing the costs and risks involved in modifying expensive, trusted, and highly specialized software.
