Definitions

Nonholonomic system

In physics and mathematics, a nonholonomic system is a system described by a set of parameters subject to differential constraints, such that when the system evolves along a path in its parameter space, where the parameters vary continuously (in the mathematical sense) and return to the identical values they held at the start of the path, the system itself may not have returned to its original state. More precisely, a nonholonomic system, also called an anholonomic system, is one in which there is a continuous closed circuit of the governing parameters, by which the system may be transformed from any given state to any other state. (reference: Geometry of Manifolds with Special Holonomy: "100 Years of Holonomy", R.L.Bryant, 150 Years of Mathematics at Washington University in St. Louis, Contemporary Mathematics, 395 (2006) pp. 29-38)

Since the final state of the system depends upon the intermediate values of its trajectory through parameter space, the system can not be represented by a conservative potential function as can, for example, the inverse square law of the gravitational force. This latter is an example of a holonomic system: path integrals in the system depend only upon the initial and final states of the system (positions in the potential), completely independent of the trajectory of transition between those states. The system is therefore said to be integrable, while the nonholonomic system is said to be nonintegrable. When a path integral is computed in a nonholonomic system, the value represents a deviation within some range of admissible values and this deviation is said to be an anholonomy produced by the specific path under consideration. This term was introduced by Heinrich Hertz in 1894. (reference: Anticipations of Geometric Phase, Michael Berry, Physics Today, December 1990, vol. 43 no. 12 pp. 34-40)

The general character of anholonomic systems is that of implicitly dependent parameters. If the implicit dependency can be removed, for example by raising the dimension of the space, thereby adding at least one additional parameter, the system is not truly nonholonomic, but is simply incompletely modeled by the lower dimensional space. In contrast, if the system intrinsically can not be represented by independent coordinates (parameters), then it is truly an anholonomic system. Some authors make much of this by creating a distinction between so-called internal and external states of the system, but in truth, all parameters are necessary to characterize the system, be they representative of "internal" or "external" processes, so the distinction is in fact artificial.

Stated more succinctly, there is a very real and irreconcilable difference between physical systems which obey conservation principles and those which do not. In the case of parallel transport on a sphere, the distinction is clear: a Riemannian manifold has a metric fundamentally distinct from that of a Euclidean space. For parallel transport on a sphere, the implicit dependence is intrinsic to the non-euclidean metric. The surface of a sphere is a two dimensional space. By raising the dimension, we can more clearly see the nature of the metric, but it is still fundamentally a two dimensional space with parameters irretrievably entwined in dependency by the Riemannian metric.

Examples

The Foucault Pendulum

The classic example of an anholonomic system is the Foucault pendulum. In the local coordinate frame the pendulum is swinging in a vertical plane with a particular orientation with respect to geographic north at the outset of the path. The implicit trajectory of the system is the line of latitude on the earth where the pendulum is located. Even though the pendulum is stationary in the earth frame, it is moving in a frame referred to the sun and rotating in synchrony with the earth's rate of revolution, so that the only apparent motion of the pendulum is that caused by the rotation of the earth. This latter frame is considered to be an inertial reference frame, although it too is non-inertial in more subtle ways. The earth frame is well known to be non-inertial a fact made perceivable by the apparent presence of centrifugal and coriolis forces.

Motion along the line of latitude is parameterized by the passage of time, and as is well known, the Foucault pendulum's plane of oscillation appears to rotate about the local vertical axis as time passes. The angle of rotation of this plane at a time t with respect to the initial orientation is the anholonomy of the system. The anholonomy induced by a complete circuit of latitude is proportional to the solid angle subtended by that circle of latitude. The path need not be constrained to latitude circles. For example, the pendulum might be mounted in an airplane. The anholonomy will still be proportional to the solid angle subtended by the path, which may now be quite irregular. The Foucault pendulum is a physical example of parallel transport.

The Rolling Sphere

This example is very easy for the reader to demonstrate. Consider a three dimensional orthogonal Cartesian coordinate frame, for example a level table top with a point marked on it for the origin, and the x and y axes laid out with pencil lines. Take a sphere of unit radius, for example a ping pong ball, and mark one point B in blue. Corresponding to this point is a diameter of the sphere, and the plane orthogonal to this diameter positioned at the center C of the sphere defines a great circle called the equator associated with point B. On this equator, select another point R and mark it in red. Position the sphere on the z=0 plane such that the point B is coincident with the origin, C is located at x=0, y=0, z=1, and R is located at x=1, y=0, and z=1, i.e. R extends in the direction of the positive x axis. This is the initial or reference orientation of the sphere.

The sphere may now be rolled along any continuous closed path in the z=0 plane, not necessarily a simply connected path, in such a way that it neither slips nor twists, so that C returns to x=0, y=0, z=1. In general, point B will no longer coincide with the origin, and point R will no longer extend along the positive x axis. In fact, by selection of a suitable path, the sphere may be re-oriented relative the initial orientation to any possible orientation of the sphere with C located at x=0, y=0, z=1. (reference: The Nonholonomy of the Rolling Sphere, Brody Dylan Johnson, The American Mathematical Monthly, June-July 2007, vol. 114, pp. 500-508) The system is therefore nonholonomic. The anholonomy may be represented by the doubly unique quaternion (q and -q) which when applied to the points representing the sphere, carries points B and R to their new positions.

Linear Polarized Light in an Optical Fiber

Take a length of optical fiber, say three meters, and lay it out in an absolutely straight line. When a vertically polarized beam is introduced at one end, it will emerge from the other end, still polarized in the vertical direction. Mark the top of the fiber with a stripe, corresponding with the orientation of the vertical polarization.

Now, coil the fiber tightly around a cylinder ten centimeters in diameter. The path of the fiber now describes a helix which, like the circle, has constant curvature. The helix also has the interesting property of having constant torsion. As such the result is a gradual rotation of the fiber about the fiber's axis as the fiber's centerline progresses along the helix. Correspondingly, the stripe will also twist about the axis of the helix.

When linearly polarized light is again introduced at one end, with the orientation of the polarization aligned with the stripe, it will, in general, emerge as linear polarized light aligned not with the stripe, but at some fixed angle to the stripe, dependent upon the length of the fiber, and the pitch and radius of the helix. This system is also nonholonomic, for we can easily coil the fiber down in a second helix and align the ends, returning the light to its point of origin. The anholonomy is therefore represented by the deviation of the angle of polarization with each circuit of the fiber. By suitable adjustment of the parameters, it is clear that any possible angular state can be produced.

Constraints

A nonholonomic constraint has the form given below and is nonintegrable:

$sum_\left\{i=1\right\}^n a_\left\{s,i\right\} d q_i + a_\left\{s,t\right\} d t = 0~~~~\left(s = 1, 2, ..., k\right)$
$n$ is the number of coordinates.
$k$ is the number of constraint equations.
$q_i$ are coordinates.
$a_\left\{s,i\right\}$ are coefficients.

In order for the above form to be nonholonomic, it is also required that the left hand side neither be a total differential nor be able to be converted into one, perhaps via an integrating factor.

For virtual displacements only, the differential form of the constraint is

$sum_\left\{i=1\right\}^n a_\left\{s,i\right\} delta q_i = 0~~~~\left(s = 1, 2, ..., k\right)$.