Editing Nonholonomic system (section)

==Constraints==
Consider a system of <math>N</math> particles with positions <math>\mathbf r_i</math> for <math>i\in\{1,\ldots,N\}</math> with respect to a given reference frame. In classical mechanics, any constraint that is not expressible as
<math display="block">f(\mathbf r_1, \mathbf r_2, \mathbf r_3, \ldots, t)=0, </math>

is a non-[[holonomic constraint]]. In other words, a nonholonomic constraint 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|p=261}} and in [[Pfaffian constraint|Pfaffian form]]:
<math display="block">\sum_{i=1}^n a_{s,i} \, dq_i + a_{s,t} \, dt = 0~~~~(s = 1, 2, \ldots, 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 |access-date=2007-09-22 |author=Jack Sarfatti |date=2000-03-26 |work=Pedagogical Review from the Classics of Physics |publisher=stardrive.org |archive-url = https://web.archive.org/web/20071020104540/http://stardrive.org/Jack/Note2.pdf <!-- Bot retrieved archive --> |archive-date = 2007-10-20}}</ref>{{rp|pp=2–3}}

For [[virtual displacement]]s only, the differential form of the constraint is<ref name="Torby1984"/>{{rp|p=282}}
<math display="block">\sum_{i=1}^n a_{s,i} \delta q_i = 0~~~~(s = 1, 2, \ldots, k).</math>

It is not necessary for all non-holonomic constraints to take this form, in fact it may involve higher derivatives or inequalities.<ref name="Herb1980">{{cite book |last=Goldstein |first=Herbert |author-link=Herbert Goldstein |title=Classical Mechanics |year=1980 |location=United States of America |publisher=Addison Wesley |edition=3rd |isbn=0-201-65702-3 |page=16}}</ref> A classical example of an inequality constraint is that of a particle placed on the surface of a sphere, yet is allowed to fall off it:
<math display="block">r^2-a^2\geq0.</math>
*<math>r</math> is the distance of the particle from the centre of the sphere.
*<math>a</math> is the radius of the sphere.