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	~~Hello~~ there, ~~I am Adrianne and I totally delve that~~ name. ~~I am~~ a ~~people manager~~ only ~~soon I~~'ll be ~~on my own~~. ~~Gardening~~ is ~~what I~~ do ~~regular~~. ~~Guam~~ has ~~always been my home~~. ~~See what~~'s ~~new~~ on this ~~website here~~: http://~~circuspartypanama~~.~~com~~<br><br>~~My page~~: [http://~~circuspartypanama~~.~~com clash~~ of ~~clans hack cydia~~]		A '''nonholonomic system''' in [[physics]] and [[mathematics]] is a [[system]] whose state depends on the path taken to achieve it. Such a system is described by a set of [[parameter]]s subject to [[differential constraint]]s, 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 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.<ref name="Bryant06">{{Cite journal\|last=Bryant\|first=R.L.\|title=Geometry of Manifolds with Special Holonomy: "100 Years of Holonomy", 150 Years of Mathematics at Washington University in St. Louis\|journal=Contemporary Mathematics\|volume=395\|year=2006\|pages=29–38}}</ref> Because the final state of the system depends on 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.<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=34–40\|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 can not 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 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 a nonholonomic 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 force]]s and [[Coriolis effect\|Coriolis]] forces.

			Motion along the line of latitude is parameterized by the passage of time, and 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 is still 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 is no longer coincident with the origin, and point R no longer extends 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 that represent 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 emerges 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 of a curve\|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 also twists 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:<ref name="Torby1984">{{cite book \|last=Torby \|first=Bruce \|title=Advanced Dynamics for Engineers \|series=HRW Series in Mechanical Engineering \|year=1984 \|publisher=CBS College Publishing \|location=United States of America \|isbn=0-03-063366-4 \|chapter=Energy Methods}}</ref>{{rp\|261}}

			:<math>\sum_{i=1}^n a_{s,i} d q_i + a_{s,t} d t = 0~~~~(s = 1, 2, ..., k)</math>
			::<math>n</math> is the number of coordinates.
			::<math>k</math> is the number of constraint equations.
			::<math>q_i</math> are coordinates.
			::<math>a_{s,i}</math> 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]].<ref name="Sarfatti2000">{{cite web \|url=http://www.stardrive.org/Jack/Note2.pdf \|title=Non Holonomic Constraints in Newtonian Mechanics \|accessdate=2007-09-22 \|author=Jack Sarfatti \|date=2000-03-26 \|format=PDF \|work=Pedagogical Review from the Classics of Physics \|publisher=stardrive.org \|archiveurl = http://web.archive.org/web/20071020104540/http://stardrive.org/Jack/Note2.pdf <!-- Bot retrieved archive --> \|archivedate = 2007-10-20}}</ref>{{rp\|2–3}}

			For [[virtual displacement]]s only, the differential form of the constraint is<ref name="Torby1984"/>{{rp\|282}}

			:<math>\sum_{i=1}^n a_{s,i} \delta q_i = 0~~~~(s = 1, 2, ..., k).</math>

			==Robotics==

			In [[robotics]] a system is non-holonomic if the controllable degrees of freedom are less than the total degrees of freedom.<ref name="flash2012">{{Cite journal\|title=Models of human movement: trajectory planning and inverse kinematics studies\|authors=Flash, Tamar ; Meirovitch, Yaron ; Barliya, Avi\|journal=Robotics and Autonomous Systems\|year=2012\|publisher=Elsevier}}</ref>
			Refer to [[holonomic (robotics)\|holonomic robotics]] for a more detailed description.

			==See also==
			*[[Bicycle and motorcycle dynamics]]
			*[[Falling cat problem]]
			*[[Holonomic (robotics)]]
			*[[Parallel parking problem]]
			*[[Pfaffian constraint]]

			==References==
			{{reflist}}

			==Variational principles for nonholonomic systems==
			*V.V. Rumiantsev, "On Hamilton's principle for nonholonomic systems" Journal of Applied Mathematics and Mechanics 42(3), (1978) 407-419.
			*[http://facta.junis.ni.ac.rs/macar/macar2000/macar2000-02.pdf V.V. Rumyantsev, "Forms of Hamilton’s principle for nonholonomic systems" Facta Universitatis. Series Mechanics, Automatic Control and Robotics 2(19), (2000) 1035-1048.]
			*V.V. Rumiantsev, "On integral principles for nonholonomic systems" Journal of Applied Mathematics and Mechanics 46(1), (1982) 1-8.

			{{DEFAULTSORT:Nonholonomic System}}
			[[Category:Algebraic topology]]
			[[Category:Differential geometry]]
			[[Category:Differential topology]]
			[[Category:Classical mechanics]]
			[[Category:Physical systems]]

182.72.79.118 at 04:54, 6 August 2012

2012-08-06T04:54:00Z

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Hello there, I am Adrianne and I totally delve that name. I am a people manager only soon I'll be on my own. Gardening is what I do regular. Guam has always been my home. See what's new on this website here: http://circuspartypanama.com<br><br>My page: [http://circuspartypanama.com clash of clans hack cydia]

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