Editing Nonholonomic system (section)

=====Additional conclusions=====
We have completed our proof that the system is nonholonomic, but our test equation gave us some insights about whether the system, if further constrained, could be holonomic. Many times test equations will return a result like <math>-1=0</math> implying the system could never be constrained to be holonomic without radically altering the system, but in our result we can see that <math>r\sin\theta</math> ''can'' be equal to zero, in two different ways:
* <math>r</math>, the radius of the wheel, can be zero. This is not helpful as the system in practice would lose all of its degrees of freedom.
* <math>\sin\theta</math> can be zero by setting <math>\theta</math> equal to zero. This implies that if the wheel were not allowed to turn and had to move only in a straight line at all times, it would be a holonomic system.
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. In this system, out of the seven additional test equations, an additional case presents itself:
<math display="block">-r\cos\theta=0</math>
This does not pose much difficulty, however, as adding the equations and dividing by <math>r</math> results in:
<math display="block">\sin\theta -\cos\theta=0</math>
which with some simple algebraic manipulation becomes:
<math display="block">\tan\theta=1</math>

which has the solution <math display="inline">\theta = \frac{\pi}{4}+n\pi;\;n\in\mathbb{Z}\;</math> (from <math>\theta = \arctan(1)</math>).

Refer back to the [[Nonholonomic system#Layman's explanation|layman's explanation]] above where it is said, "[The valve stem's] new position depends on the path taken. If the wheel were holonomic, then the valve stem would always end up in the same position as long as the wheel were always rolled back to the same location on the Earth. Clearly, however, this is not the case, so the system is nonholonomic." 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! In fact, moving parallel to the given angle of <math>\pi/4</math> is not actually necessary in the real world as the orientation of the coordinate system itself is arbitrary. The system can become holonomic if the wheel moves only in a straight line at any fixed angle relative to a given reference. Thus, we have not only proved that the original system is nonholonomic, but we also were able to find a restriction that can be added to the system to make it holonomic.

However, there is something mathematically special about the restriction of <math>\theta = \arctan(1)</math> for the system to make it holonomic, as <math>\theta = \arctan(y/x)</math> in a Cartesian grid. Combining the two equations and eliminating <math>\theta</math>, we indeed see that <math>y = x</math> and therefore one of those two coordinates is completely redundant. 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 <math> \mathbf{u}=\begin{bmatrix} x & \phi \end{bmatrix}^\mathrm{T} </math>. As discussed [[Holonomic constraints#Configuration spaces of two or one variable|here]], a system that is modellable by a Pfaffian constraint must be holonomic if the configuration space consists of two or fewer variables. By modifying our original system to restrict it to have only two degrees of freedom and thus requiring only two variables to be described, and assuming it can be described in Pfaffian form (which in this example we already know is true), we are assured that it is holonomic.