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. N. M. Ferrers first suggested to extend the equations of motion with nonholonomic constraints in 1871.He introduced the expressions for Cartesian velocities in terms of generalized velocities.In 1877, E. Routh wrote the equations with the Lagrange multipliers. In the third edition of his book for linear non-holonomic constraints of rigid bodies, he introduced the form with multipliers, which is now called the Lagrange equations of the second kind with multipliers. The terms the holonomic and nonholonomic systems were introduced by Heinrich Hertz in 1894.In 1897, S. A. Chaplygin first suggested to form the equations of motion without Lagrange multipliers.Under certain linear equations of constraint, he discriminated in the left-hand side of equations of motion the group of extra terms of the type of the Lagrange operator.The remaining extra terms characterize the nonholonomicity of system and they go to zeros when the given constrains are integrable.In 1901 P. V.Voronets generalized Chaplygin's work to the cases of noncyclic holonomic coordinates and of nonstationary constraints. Consider a system of N {displaystyle N} particles with positions r i {displaystyle mathbf {r} _{i}} for i ∈ { 1 , … , N } {displaystyle iin {1,ldots ,N}} with respect to a given reference frame. In classical mechanics, any constraint that is not expressible as is a non-holonomic constraint. In other words, a nonholonomic constraint is nonintegrable:261 and has the form 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.:2–3 For virtual displacements only, the differential form of the constraint is:282