I've posted this on Math Stack Exchange and I didn't get any answer in a couple of days, so I'll try and post it here too.

The problem presented below is from my differential geometry course. The initial reference is Nelson, Tensor Analysis 1967. The car is modelled as follows:

Denote by $C(x,y)$ the center of the back wheel line, $\theta$ the angle of the direction of the car with the horizontal direction, $\phi$ the angle made by the front wheels with the direction of the car and $L$ the length of the car.

The possible movements of the car are denoted as follows:

• steering: $S=\displaystyle\frac{\partial}{\partial \phi}$;
• drive: $D=\displaystyle\cos \theta \frac{\partial}{\partial x}+\sin\theta \frac{\partial}{\partial y}+\frac{\tan \phi}{L}\frac{\partial}{\partial \theta}$;
• rotation: $R=[S,D]=\displaystyle\frac{1}{L\cos^2 \phi}\frac{\partial }{\partial \theta}$;
• translation: $T=[R,D]=\displaystyle\frac{\cos \theta}{L\cos^2 \phi}\frac{\partial}{\partial y}-\frac{\sin\theta}{L\cos^2\phi}\frac{\partial}{\partial x}$

Where $[X,Y]=XY-YX$ (I can't remember the English word now). All these transformations seem very logical. My question is:

How can we justify the mathematical interpretation made above, especially the part with the rotations and translations?

The interpretations are quite interesting:

• from the expression of $D$, when the car is shorter, you can change the orientation of the car very easily, but when it is longer, like a truck, you it is not that easy ( see the term with $\frac{\partial}{\partial \theta}$)
• the rotation is faster for smaller cars, and for greater steering angle
• translation is easier for smaller cars.