01 Mechanics and Nonlinear Dynamics

The subject of mechanics is perhaps the most apparent and everyday aspect of physics. From early childhood, our brains learn the practical physics of mechanical motion to crawl, walk, run and eventually perform remarkably agile feats - even, for example, the manipulation of food and utensils with hands and fingers. How do we translate our experience of forces and movement into a quantitative and broadly predictive physical theory? What in fact is force and how do we relate it to concepts like momentum when, for example, observing the collision of two football players or the launch of a rocket? Once we have a grasp of modeling systems in terms of force, momentum, and motion, how can we exploit this to achieve precise control over systems as widely varied as loads swinging below cranes and the in-flight maneuvers of supersonic aircraft? Can we bring the elegant Lagrangian formulation of mechanics into service in these applications or in, say, robotics? We experience that friction ultimately brings things to a halt and yet ticking mechanical clocks and the beating of our hearts tells us that somehow we and nature can contrive to maintain a state of constant periodic motion. What principles underly the spontaneous onset of such self-sustained oscillation? What new complexities must we entertain when dealing with extended objects held together cohesively by internal forces when these objects interact with external forces? How are the external forces in fact transmitted through real solid bodies and when are they likely to produce catastrophic failure of a material? What is the effect of external forces when the solid objects are in a state of rotation? Sometimes the effects can be unexpected and subtle so that even now, after Newton's investigations 350 years ago, we discover (and sometime re-discover) remarkable behavior. The precession and wobbling of rotating top is just one example. More exotic is the unpredictable behavior called chaos that arises for remarkably simple systems like a pendulum or a bouncing ball subjected to regular external perturbation. And the archetype of all predictability, the motion of the earth in its orbit around the sun and the moon in orbit around the earth, belies a more exotic picture of chaos and even orbital ejection that can arise when three or more bodies interact through gravity. How can we bring such exotic behavior more into our awareness and understand deeply how fortunate we are, or any living civilization might be, to arise in a universe in which there is an ongoing interplay between regularity and chaos built into the laws of physics.