Nonholonomic system

A nonholonomic system in physics and mathematics is a system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differential constraints, such that when the system evolves along a path in its parameter space (the parameters varying continuously in values) but finally returns to the original set of parameter values 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. Because the final state of the system depends on the intermediate values of its trajectory through parameter space, the system cannot 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.

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 cannot 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. However, there is a very real and irreconcilable difference between physical systems that obey conservation principles and those that 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.

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