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

• by m <n conditions ( constraints )

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

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.

See also

• Holonomy
• Holonomic constraints