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

enter image description here

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.
  • 1
    $\begingroup$ I believe the word you are looking for is commutator :) $\endgroup$
    – Tyler
    May 26, 2011 at 21:10
  • $\begingroup$ No one can explain this? Should I move it to MathOverflow? $\endgroup$ May 30, 2011 at 8:08
  • 1
    $\begingroup$ I see you did, and got a nice answer mathoverflow.net/questions/66578/… $\endgroup$
    – yasmar
    Jun 2, 2011 at 20:42

1 Answer 1


I asked the question on MathOverflow, and got a very nice answer. You can see it here.

  • $\begingroup$ Oops, now I see that in the comment above yasmar has already posted that. Sorry about the double post... $\endgroup$ Jun 7, 2011 at 8:13
  • $\begingroup$ hey! Beni, are you're self promoting? :D don't worry the subject is worth it!! $\endgroup$
    – janmarqz
    Feb 13, 2014 at 1:57
  • $\begingroup$ @janmarqz: I don't get it. What self promoting are you talking about? $\endgroup$ Feb 13, 2014 at 17:59
  • $\begingroup$ you ask, you answer, you hyperlink... etc, any way there's no problem $\endgroup$
    – janmarqz
    Feb 13, 2014 at 19:31
  • $\begingroup$ It is not my answer I link to. When the question was unanswered here, I cross-posted it to MathOverflow. $\endgroup$ Feb 13, 2014 at 23:36

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