Geometric Mechanics, Stability and Control

This paper gives an overview of selected topics in mechanics and their relation to questions of stability, control and stabilization. The mechanical connection, whose holonomy gives phases and that plays an important role in block diagonalization, provides a unifying theme. 1 The Mechanical Connection The mechanical connection is a basic object that links mechanics and gauge theory. This point of view has matured over the last decade to the stage where we now see enough applications and intrinsic beauty to elevate it to a special status. This topic owes much to Smale [1970], Meyer [1973], Marsden and Weinstein [1974], Abraham and Marsden [1978], Kummer [1981], Guichardet [1984], Montgomery, Marsden and Ratiu [1984], Montgomery [1986,1989,1990], Iwai [1987], Wilczeck and Shapere [1989], Marsden, Montgomery and Ratiu [1990], Simo, Posbergh and Marsden [1990], Simo, Lewis and Marsden [1991] and others. We start with four basic ingredients: • Q, the configuration manifold of a mechanical system, • 〈〈 , 〉〉, a Riemannian metric (usually derived from the kinetic energy), • G, a Lie group of symmetries acting by isometries on Q, • V , a G-invariant potential energy. ∗Research partially supported by a Fairchild fellowship at Caltech and by DOE. We thank W. Sluis for providing a draft of the lectures given at the Fields Institute, March 1992.