Editing Nonholonomic system (section)

==Details==
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.<ref name="Bryant06">{{cite conference
 | last = Bryant | first = Robert L.
 | contribution = Geometry of manifolds with special holonomy: '100 years of holonomy'
 | doi = 10.1090/conm/395/07414
 | mr = 2206889
 | pages = 29–38
 | publisher = American Mathematical Society | location = Providence, RI
 | series = Contemporary Mathematics
 | title = 150 years of mathematics at Washington University in St. Louis
 | volume = 395
 | year = 2006| doi-access = free
 }}</ref> 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 [[Scalar potential|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.<ref name="Berry90">{{Cite journal|title=Anticipations of Geometric Phase|last=Berry|first=Michael|journal=Physics Today|date=December 1990|volume=43|issue=12|pages=34&ndash;40|doi=10.1063/1.881219|bibcode = 1990PhT....43l..34B }}</ref>

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{{Citation needed|date=January 2010}} 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 tensor|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{{clarify|reason=I certainly can't. Could someone knowledgeable please demystify this paragraph?|date=April 2017}} the nature of the metric, but it is still fundamentally a two-dimensional space with parameters irretrievably entwined in dependency by the [[Riemannian metric]].

By contrast, one can consider an X-Y [[plotter]] as an example of a '''holonomic''' system where the state of the system's mechanical components will have a single fixed configuration for any given position of the plotter pen. If the pen relocates between positions 0,0 and 3,3, the mechanism's gears will have the same final positions regardless of whether the relocation happens by the mechanism first incrementing 3 units on the x-axis and then 3 units on the y-axis, incrementing the Y-axis position first, or operating any other sequence of position-changes that result in a final position of 3,3. Since the final state of the machine is the same regardless of the path taken by the plotter-pen to get to its new position, the end result can be said not to be ''path-dependent''. If we substitute a [[Turtle (robot)|turtle]] plotter, the process of moving the pen from 0,0 to 3,3 can result in the gears of the robot's mechanism finishing in different positions depending on the path taken to move between the two positions. See this very similar [[Holonomic constraints#Gantry crane|gantry crane]] example for a mathematical explanation of why such a system is holonomic.