is the angular version of the familiar relation ''momentum equals mass times velocity.'' Shape changes result in a change in the cat's moment of inertia and this, together with the constancy of the angular momentum, creates the overall orientation change. However, the exact process by which this occurs is subtle, and intuitive reasoning can lead one astray. While this problem has been long studied (e.g., by Kane and Shur, 1969), recently new and interesting insights have been discovered using geometric methods (see Enos, 1993; Montgomery, 1990, and references therein).
Astronauts who wish to reorient themselves in a free space environment can similarly do so by means of shape changes. For example, holding one of their legs straight, they can swivel it at the hip, moving their foot in a circle. When they have achieved the desired orientation, they merely stop their leg movement. Similar movements for robots and spacecraft can be controlled automatically to achieve desired objectives (see, for example, Walsh and Sastry, 1995). One often refers to the extra motion that is achieved as the geometric phase.
The history of this phenomenon and its applications is a long and complex story. We shall only mention a few highlights. Certainly the shift in the plane of the swing in the Foucault pendulum as the earth rotates once around its axis is one of the earliest examples of this phenomenon. Anomalous spectral shifts in rotating molecules are another. Phase formulas for special problems such as rigid body motion and polarized light in helical fibers were understood already in the early 1950s. Additional historical comments and references can be found in Berry (1990), and Marsden and Ratiu (1994). Gradually the subject became better understood, but the first paper to clarify and emphasize the ubiquity of the geometry behind all these phenomena was Berry (1985). It was also quickly realized that the phenomenon occurs in essentially the same way in both classical and quantum mechanics (Hannay, 1985), and that the phenomenon can be linked in a fundamental way with the presence of symmetry (Montgomery, 1988; Marsden et al., 1990).
The theory of geometric phases has an interesting link with noneuclidean geometry, a subject first invented for its own sake, without regard to applications. A simple way to explain this link is as follows. Hold your hand at arm's length, but allow rotation in your shoulder joint. Move your hand along three great circles, forming a triangle on the sphere, and during the motion, keep your thumb "parallel," that is, forming a fixed angle with the direction of motion. After completing the circuit around the triangle, your thumb will return rotated through an angle relative to its starting position (see Figure 1.1). In fact, this angle (in radians) is given by Θ = Δ-π where Δ is the sum of the angles of the triangle. The fact that is of course one of the basic facts of noneuclidean geometry—in curved spaces, the sum of the angles of a triangle is not necessarily z (i.e., 180º). This angle is also related to the area A enclosed by the triangle through the relation Θ = A/r2, where r is the radius of the sphere.
The examples presented so far are rather different from what one finds in many other mechanical systems of interest in one crucial aspect—the absence of constraints of rolling, sliding, or contact. For example, when one parks a car, the steering mechanism is manipulated and movement into the parking spot is generated; obviously the rolling of the wheels on the road is crucial to the maneuver. When a human or a robot manipulates an object in its fingers (imagine twirling an egg in your fingers), it can reorient the object through the rolling of its fingers on the object. This can be shown in a demonstration I learned from Roger Brockett: roll your fingers in a rotating motion on a ball resting on a table—you will find that the ball reorients itself under your finger! The amount of rotation is again related to the amount of area you capture in the rotating motion. You have generated rotational motion! (See Figure 1.2.)