Holonomic constraints
In classical mechanics, holonomic constraints are relations between the position variables (and possibly time)[1] that can be expressed in the following form:
are n generalized coordinates that describe the system (in unconstrained configuration space).
For example, the motion of a particle constrained to lie on the surface of a sphere is subject to a holonomic constraint, but if the particle is able to fall off the sphere under the influence of gravity, the constraint becomes non-holonomic.
Examples of holonomic systems are gantry cranes, pendulums, and robotic arms.
Examples of nonholonomic systems are Segways, unicycles, and automobiles.
lists the displacement of the components of the system, one for each degree of freedom.
A system that can be described using a configuration space is called scleronomic.
For example, the total allowable motion of a pendulum can be described with a scleronomic constraint, but the motion over time of a pendulum must be described with a rheonomic constraint.
If the origin of the coordinate system is at the back-bottom-left of the crane, then we can write the position constraint equation as:
Optionally, we may simplify to the standard form where all constants are placed after the variables:
As shown on the right, a simple pendulum is a system composed of a weight and a string.
The particles of a rigid body obey the holonomic constraint
If a given system is holonomic, rigidly attaching additional parts to components of the system in question cannot make it non-holonomic, assuming that the degrees of freedom are not reduced (in other words, assuming the configuration space is unchanged).
Therefore, all holonomic and some nonholonomic constraints can be expressed using the differential form.
Examples of nonholonomic constraints that cannot be expressed this way are those that are dependent on generalized velocities.
sets of constraint equations (note that variable(s) representing time can be included, as from above
Consider this dynamical system described by a constraint equation in Pfaffian form.
Because there are only three terms in the configuration space, there will be only one test equation needed.
We can organize the terms of the constraint equation as such, in preparation for substitution:
It's easy to see that we can combine the results of our integrations to find the holonomic constraint equation:
We may prove this as follows: consider a system of constraints in Pfaffian form where every coefficient of every differential is a constant, as described directly above.
It is well-known in calculus that any derivative (full or partial) of any constant is
However, the universal test requires three variables in the configuration or state space.
is by definition not a measure of anything in the system, its coefficient in the Pfaffian form must be
A similar proof can be conducted with one actual variable in the configuration or state space and two dummy variables to confirm that one-degree-of-freedom systems describable in Pfaffian form are also always holonomic.
In conclusion, we realize that even though it is possible to model nonholonomic systems in Pfaffian form, any system modellable in Pfaffian form with two or fewer degrees of freedom (the number of degrees of freedom is equal to the number of terms in the configuration space) must be holonomic.
Important note: realize that the test equation failed because the dummy variable, and hence the dummy differential included in the test, will differentiate anything that is a function of the actual configuration or state space variables to
, there are still three degrees of freedom described in the configuration or state space.
The holonomic constraint equations can help us easily remove some of the dependent variables in our system.
In order to study classical physics rigorously and methodically, we need to classify systems.
