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); these conditions are assumed to be expressible as non-integrable differential relations

$$ \tag{1 } \phi _ {s} ( x _ {1} \dots x _ {3N} ,\ \dot{x} _ {1} \dots \dot{x} _ {3N} , t) = 0, $$

$$ s = 1 \dots m,\ \phi _ {s} ( x, \dot{x} , t) \in C ^ {1} , $$

that cannot be replaced by equivalent finite relationships among the coordinates. Here, the $ x _ \nu $ denote the Cartesian coordinates of the points, $ t $ is the time and $ N $ is the number of points in the system. Most often one considers constraints (1) that are linear in the velocities $ \dot{x} _ {i} $, of the form

$$ \sum _ {i = 1 } ^ { 3N } A _ {s _ {i} } dx _ {i} + A _ {s} dt = 0; \ \ A _ {s _ {i} } ( x, t),\ A _ {s} ( x, t) \in C ^ {1} . $$

The constraints (1) are said to be stationary if $ \partial \phi / \partial t \equiv 0 $. These constraints also impose conditions on the accelerations $ w _ \nu $ of the points:

$$ \frac{\partial \phi _ {s} }{\partial t } = \ \sum _ {\nu = 1 } ^ { N } \mathop{\rm grad} _ {\dot{r} _ \nu } \ \phi _ {s} \cdot w _ \nu + \dots = 0. $$

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

$$ \tag{2 } \sum _ {\nu = 1 } ^ { 3N } \frac{\partial \phi _ {s} }{\partial \dot{x} _ \nu } \delta x _ \nu = 0,\ \ s = 1 \dots m. $$

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

$$ \sum _ {i = 1 } ^ { 3N } A _ {s _ {i} } \delta x _ {i} = 0. $$

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 [1]).

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

$$ \Phi _ {s} ( q _ {1} \dots q _ {n} ,\ \dot{q} _ {1} \dots \dot{q} _ {n} , t) = 0,\ \ \sum _ {i = 1 } ^ { n } \frac{\partial \Phi _ {s} }{\partial \dot{q} _ {i} } \delta q _ {i} = 0, $$

$$ s = 1 \dots m. $$

In a non-holonomic system, the number $ n - m $ of degrees of freedom is less than the number $ n $ of independent coordinates $ q _ {i} $ by the number $ m $ 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)), the Appell equations in Lagrange coordinates and quasi-coordinates, the Chaplygin–Voronets equations in Lagrange coordinates, the Boltzmann equation, the Hamel equation in quasi-coordinates, etc. (see [3]).

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

References

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

References

[a1]	L.D. Landau, E.M. Lifshitz, "Course of theoretical physics" , 1. Mechanics , Pergamon (1976)
[a2]	E.G.G. Sudarshan, "Classical dynamics: a modern perspective" , Wiley (Interscience) (1974)