Holonome

Holonom ( Greek : “completely legal”) is a property of a mechanical system. A holonomic system of bodies is characterized by the fact that the position of the body can be described by generalized coordinates , which ${\ displaystyle n}$ ${\ displaystyle q_ {1}, q_ {2}, \ ldots, q_ {n}}$

• are completely independent of each other

or

${\ displaystyle a_ {i} (q_ {1}, q_ {2} ... q_ {n}, t) = 0; \ quad i \ in [1, m]}$
are connected.

How many generalized coordinates describe the system, i.e. what numerical value the index has, must be determined by determining the degrees of freedom of the system. ${\ displaystyle n}$

Non-holonomic systems

If at least one of the conditions contains one or more speed coordinates (time derivative of the generalized coordinates), it is of the form ${\ displaystyle a_ {i}}$${\ displaystyle {\ dot {q}}}$

${\ displaystyle a_ {i} (q_ {1}, q_ {2} ... q_ {n}, {\ dot {q}} _ {1}, {\ dot {q}} _ {2} .. . {\ dot {q}} _ {n}, t) = 0,}$

and if the speed coordinates cannot be eliminated by integration , the system is non-holonomic .

Wheel rolling on the xy plane (top view)

As an example, the wheel of a vehicle rolls without sliding on a flat surface. The independence of the coordinates , and is restricted by the non-integrable condition ${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle \ varphi}$

${\ displaystyle v = - {\ frac {\ dot {x}} {\ sin \ varphi}} = {\ frac {\ dot {y}} {\ cos \ varphi}}}$

${\ displaystyle \ Leftrightarrow {\ dot {x}} \ cdot \ cos \ varphi + {\ dot {y}} \ cdot \ sin \ varphi = 0.}$

d. H. the direction of the rolling movement can only be perpendicular to the wheel axis. ${\ displaystyle {\ vec {v}}}$

While every constellation of the system with the arbitrarily chosen coordinates , and is permissible (3  degrees of freedom "on a large scale"), so the wheel can take any position and orientation in the plane, there is a restriction when transitioning from one constellation to an infinitesimally neighboring one by the above non-holonomic roll condition; “On a small scale” there are therefore only two degrees of freedom. ${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle \ varphi}$

This fact becomes even clearer when the facts are transferred to a four-wheeled vehicle with front-wheel steering : Even if a parking space offers enough space for the vehicle, it may be impossible to get into it.