Nonholonomic system
[2] Because the final state of the system depends on the intermediate values of its trajectory through parameter space, the system cannot be represented by a conservative potential function as can, for example, the inverse square law of the gravitational force.
When a path integral is computed in a nonholonomic system, the value represents a deviation within some range of admissible values and this deviation is said to be an anholonomy produced by the specific path under consideration.
Some authors[citation needed] make much of this by creating a distinction between so-called internal and external states of the system, but in truth, all parameters are necessary to characterize the system, be they representative of "internal" or "external" processes, so the distinction is in fact artificial.
However, there is a very real and irreconcilable difference between physical systems that obey conservation principles and those that do not.
For parallel transport on a sphere, the implicit dependence is intrinsic to the non-euclidean metric.
Since the final state of the machine is the same regardless of the path taken by the plotter-pen to get to its new position, the end result can be said not to be path-dependent.
If we substitute a turtle plotter, the process of moving the pen from 0,0 to 3,3 can result in the gears of the robot's mechanism finishing in different positions depending on the path taken to move between the two positions.
N. M. Ferrers first suggested to extend the equations of motion with nonholonomic constraints in 1871.
[7] Under certain linear constraints, he introduced on the left-hand side of the equations of motion a group of extra terms of the Lagrange-operator type.
The remaining extra terms characterise the nonholonomicity of system and they become zero when the given constrains are integrable.
In 1901 P. V.Voronets generalised Chaplygin's work to the cases of noncyclic holonomic coordinates and of nonstationary constraints.
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.
It is not necessary for all non-holonomic constraints to take this form, in fact it may involve higher derivatives or inequalities.
[11] 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:
Now we do some algebraic manipulation to transform the equation to Pfaffian form so it is possible to test whether it is holonomic, starting with:
Therefore, it is often best practice to have the first test equation have as many non-zero terms as possible to maximize the chance of the sum of them not equaling zero.
can be equal to zero, in two different ways: There is one thing that we have not yet considered however, that to find all such modifications for a system, one must perform all eight test equations (four from each constraint equation) and collect all the failures to gather all requirements to make the system holonomic, if possible.
Refer back to the layman's explanation above where it is said, "[The valve stem's] new position depends on the path taken.
However it is easy to visualize that if the wheel were only allowed to roll in a perfectly straight line and back, the valve stem would end up in the same position!
The system can become holonomic if the wheel moves only in a straight line at any fixed angle relative to a given reference.
We already know that the steering angle is a constant, so that means the holonomic system here needs to only have a configuration space of
As discussed here, a system that is modellable by a Pfaffian constraint must be holonomic if the configuration space consists of two or fewer variables.
Consider a three-dimensional orthogonal Cartesian coordinate frame, for example, a level table top with a point marked on it for the origin, and the x and y axes laid out with pencil lines.
Take a sphere of unit radius, for example, a ping-pong ball, and mark one point B in blue.
In the local coordinate frame the pendulum is swinging in a vertical plane with a particular orientation with respect to geographic north at the outset of the path.
The implicit trajectory of the system is the line of latitude on the Earth where the pendulum is located.
The angle of rotation of this plane at a time t with respect to the initial orientation is the anholonomy of the system.
Take a length of optical fiber, say three meters, and lay it out in an absolutely straight line.
Mark the top of the fiber with a stripe, corresponding with the orientation of the vertical polarization.
This system is also nonholonomic, for we can easily coil the fiber down in a second helix and align the ends, returning the light to its point of origin.
