# Symmetry, Stability, Geometric Phases, and Mechanical Integrators

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## Abstract

New analytical techniques and recent algorithms which numerically compute the time evolution of mechanical systems enable today's scientists. engineers. and mathematicians to predict events more accurately and more rapidly than ever before. Beyond the problems of simulation and prediction, however, tie the problems of understanding a dynamical system and choosing a correct dynamical system to model a given physical situation. Many systems remain too intricate to fully understand, but modern methods of mathematical analysis can sometimes offer insight. Most of this insight is obtained by viewing dynamics geometrically, and in fact the recent advances in mechanics which we review in this article all share this geometric perspective. Much of the value of these techniques lies in their applications, and although applications exist in a broad range of dlsciplines, we will focus on examples from space mechanics and robotics because these are simple to visualize. A key problem in space mechanics is the problem of efficiently and effectively controlling the attitude of satellites in their orbits. Several spacecraft. including the very first U.S. satellite, Explorer I. have been unable to complete their missions because they began to tumble in space and could not be stabilized. Much research has been devoted to prevent current orbiting telescopes from suffering a similar fate. These telescopes must be controlled with high precision. since small errors can seriously degrade observations made of objects thousands of light years away. Several problems have plagued the Hubble Space Telescope. including low-frequency vibrations in the structure's solar-power panels due to unanticipated thermal expansion effects as the telescope passes from night into day. These vibrations were further amplified by the telescope's computer controlled stabilization mechanisms (Wilford, [1990]). Two of the topics we shall discuss - stability and numerical integration - are pertinent to the analysis and control of such vibrations. Stability and control are also important issues in the field of robotics. This is certainly the case for a team at the MIT Artificial Intelligence Laboratory which is trying to construct a somersaulting robot (Hodgins and Raibert [1989]) as shown in Figure I. Specifically. the project is to build a robot which will gather a running start, launch itself into the air, complete a forward revolution, and then land firmly on its feet. As might be imagined, the challenges involved in such a venture are formidable. Recent ideas of Berry [1984, 1985], Hannay [1985], and Montgomery [1990], however, may help to solve this problem as well as provide the means for a way of efficiently controlling mechanical systems such as orbiting telescopes. It is amusing to note that many of these recent ideas are related to a natural curiosity that has fascinated and motivated investigations in physiology as well as dynamics: How does a falling cat often manage to land upright even if released while upside down from a complete rest? (See Figure 2.) The cat cannot violate the conservation of angular momentum, yet somehow it manages to tum itself 180 degrees in mid-air. This process has been investigated many times over the past century (see Nature 1 I 894), Crabtree [1909], Kane and Scher [1969] and references therein) and recently has been analyzed by Montgomery [1990] with an emphasis on how the cat (or, more generally, a deformable body) can efficiently readjust its orientation by changing its shape. By "efficiently," we mean that the reorientation minimizes some function - for example the total energy expended. Montgomery's results characterize the deformations which allow a cat to most efficiently reorient itself without violating conservation of angular momentum. We begin with a review of Hamiltonian systems and canonical formulations. We then introduce noncanonical formulations and the concept of reduction of dynamics. Recent results in determining stability arc presented in the next section, and these are followed by a discussion of geometric phases in mechanics. We conclude with a survey of some recent advances in numerical integration algorithms.

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