View source for Holonomic system

A system of material points that is either not constrained by any constraint or constrained only by geometric constraints. The latter impose restrictions on the positions of the points of the system and may be represented by relations 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/h/h047/h047580/h0475801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047580/h0475802.png" /> is the time, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047580/h0475803.png" /> are the Cartesian coordinates of the point and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047580/h0475804.png" /> is the number of points in the system. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047580/h0475805.png" />, the constraints are said to be stationary; otherwise they are called non-stationary. Any position of the system for which the coordinates of the points obey equations (1) is called possible for the given moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047580/h0475806.png" />. The constraints (1) impose restrictions not only on the positions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047580/h0475807.png" />, but also on the velocities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047580/h0475808.png" /> and on the accelerations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047580/h0475809.png" /> of the points:

<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/h/h047/h047580/h04758010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

The velocities and accelerations satisfying equations (2) are said to be kinematically possible in a given position <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047580/h04758011.png" /> of the system for a given moment of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047580/h04758012.png" />. Infinitesimal displacements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047580/h04758013.png" /> which 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/h/h047/h047580/h04758014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

are the possible (virtual) displacements of the system, as distinct from the real displacements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047580/h04758015.png" /> performed by the system within the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047580/h04758016.png" /> under the effect of forces applied to it and satisfying 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/h/h047/h047580/h04758017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

For stationary constraints real displacements are found among the possible displacements, while for non-stationary constraints they are not found among the possible displacements, in general. Possible displacements are useful for converting a holonomic system from one position of the system which is possible for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047580/h04758018.png" /> to another infinitely-close position which is possible at the same moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047580/h04758019.png" />.

The number of independent variations of the points of the system is said to be the number of its degrees of freedom; for a holonomic system this coincides with the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047580/h04758020.png" /> of independent arbitrary parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047580/h04758021.png" /> with the aid of which equations (1) may be represented in the form of relations 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/h/h047/h047580/h04758022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

The parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047580/h04758023.png" /> are called generalized, or Lagrangian, coordinates of the system; they are called holonomic coordinates, in distinction from non-holonomic, or quasi-, coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047580/h04758024.png" />, introduced by non-integrable relations 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/h/h047/h047580/h04758025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

Constraints that can be analytically expressed by (1) are said to be retaining, or two-sided, in distinction from non-retaining, or one-sided, constraints, which can be expressed by inequalities 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/h/h047/h047580/h04758026.png" /></td> </tr></table>

and impose the following conditions on the possible displacements:

<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/h/h047/h047580/h04758027.png" /></td> </tr></table>

The possible displacements of a system with two-sided constraints are reversible; among the possible displacements of a system with one-sided constraints there are irreversible ones [[#References|[1]]].

The motions of holonomic systems are described by the [[Lagrange equations (in mechanics)|Lagrange equations (in mechanics)]] (of the first and second kinds), by the [[Hamilton equations|Hamilton equations]] in Lagrangian coordinates and impulses, the [[Appell equations|Appell equations]], the Poincaré equations or the [[Chetaev equations|Chetaev equations]] in Lagrangian coordinates and quasi-coordinates.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.K. Suslov,   "Theoretical mechanics" , Moscow  (1944)  (In Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.E. Rutherford,   "Classical mechanics" , Oliver &amp; Boyd  (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Th.T. Taylor,   "Mechanics: classical and quantum" , Pergamon  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.D. Landau,   E.M. Lifshitz,   "A course of theoretical physics" , '''1. Mechanics''' , Pergamon  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.A. Hylleraas,   "Mathematical and theoretical physics" , '''1''' , Wiley (Interscience)  (1970)</TD></TR></table>