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I've come accross this physical interpretation for $ [X,Y] $ which I don't understand :

  • Follow $X$ for some time $\epsilon$;
  • Follow $Y$ for $\epsilon$;
  • Follow -X for $\epsilon$;
  • Follow -Y for $\epsilon$;

In the limit as $\epsilon$ approaches 0, the result of the above motion approaches the Lie Bracket $[X,Y]$.

Maybe someone can elucidate this for me?

  • 0
    [This](http://math.stackexchange.com/questions/163262/visualizing-commutator-of-two-vector-fields) might be related..2012-07-08
  • 8
    A note on magnitudes might be in order: When $\epsilon$ is small, the above procedure approximates following $[X,Y]$ for $\epsilon^2$. There is a hilarious example in one of Edward Nelsons lecture notes from around 1970 (I think), using double commutators to show that you can drive a car sideways, at least to within any desired tolerance. But the $\epsilon^2$ factor makes this method of parallel parking exceedingly laborious.2012-07-08
  • 0
    As Harald mentions, the Lie Bracket is the 2nd derivative of the function of $\epsilon$ you describe in your question, evaluated at $\epsilon = 0$.2012-08-07
  • 0
    @HaraldHanche-Olsen I think you mean the example starting on page 33 in [Nelson's 1967 notes](https://web.math.princeton.edu/~nelson/books/ta.pdf) on tensor analysis. Fantastic! I only knew a simplified version of this example from lectures by Salamon, see also example 1.68 on page 34 (42 of the pdf) in [these notes](http://www.math.ethz.ch/%7Esalamon/PREPRINTS/diffgeo2011.pdf) by Salamon-Robbin.2012-09-06
  • 0
    @t.b. That's the one!2012-09-07

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