View source for Non-holonomic systems

Systems of material points that are subject to constraints among which are kinematic constraints that impose conditions on the velocities (and not only the positions) of the points of the system in its possible positions (see [[Holonomic system|Holonomic system]]); these conditions are assumed to be expressible as non-integrable differential relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067050/n0670501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

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that cannot be replaced by equivalent finite relationships among the coordinates. Here, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067050/n0670503.png" /> denote the Cartesian coordinates of the points, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067050/n0670504.png" /> is the time and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067050/n0670505.png" /> is the number of points in the system. Most often one considers constraints (1) that are linear in the velocities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067050/n0670506.png" />, of the form

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The constraints (1) are said to be stationary if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067050/n0670508.png" />. These constraints also impose conditions on the accelerations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067050/n0670509.png" /> of the points:

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Following N.G. Chetaev [[#References|[2]]], assume that the possible motions of the systems subject to the non-linear constraints (1) satisfy conditions of the type

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067050/n06705011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

In the case of linear constraints, these conditions imply the usual relations

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Unlike the situation in holonomic systems, motion between neighbouring positions at an infinitesimally-small distance from one another may be impossible in a non-holonomic system (see [[#References|[1]]]).

In generalized Lagrange coordinates, equations (1) and (2) are written as

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067050/n06705014.png" /></td> </tr></table>

In a non-holonomic system, the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067050/n06705015.png" /> of degrees of freedom is less than the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067050/n06705016.png" /> of independent coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067050/n06705017.png" /> by the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067050/n06705018.png" /> of non-integrable constraint equations.

Many and varied forms of differential equations of motion have been derived for non-holonomic systems, such as the Lagrange equation of the first kind (cf. [[Lagrange equations (in mechanics)|Lagrange equations (in mechanics)]]), the [[Appell equations|Appell equations]] in Lagrange coordinates and quasi-coordinates, the Chaplygin–Voronets equations in Lagrange coordinates, the [[Boltzmann equation|Boltzmann equation]], the Hamel equation in quasi-coordinates, etc. (see [[#References|[3]]]).

A characteristic feature of non-holonomic systems is that, in the general case, their differential equations of motion include the constraint equations.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Hertz,   "The principles of mechanics presented in a new form" , Dover  (1956)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.G. Chetaev,   ''Izv. Fiz.-Mat. Obshch. Kazan. Univ. (3)'' , '''6'''  (1932)  pp. 68–71</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Yu.I. Neimark,   N.A. Fufaev,   "Dynamics of nonholonomic systems" , Amer. Math. Soc.  (1972)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.D. Landau,   E.M. Lifshitz,   "Course of theoretical physics" , '''1. Mechanics''' , Pergamon  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.G.G. Sudarshan,   "Classical dynamics: a modern perspective" , Wiley (Interscience)  (1974)</TD></TR></table>