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A Definition of the Lie derivative

  1. Oct 27, 2016 #1
    Consider the Lie derivative of the vector field ##\bf{Y}## with respect to the vector field ##\bf{X}## on manifold ##M^{n}(x)## defined as

    ##\displaystyle{[\mathcal{L}_{\bf{X}}Y]_{x}:=\lim_{t\rightarrow 0} \frac{[{\bf{Y}}_{\phi_{t}x}-\phi_{t*}{\bf{Y}}_{x}]}{t}}##

    Now, I understand that ##{\bf{Y}}_{\phi_{t}x}## is the tangent vector of the vector field ##\bf{Y}## at the point ##\phi_{t}x##, where the point ##\phi_{t}x## is obtained by starting at point ##x## at time ##0## and traversing along the orbit of ##x## to time ##t##.

    But I don't understand how to interpret ##\phi_{t*}{\bf{Y}}_{x}##. Given the map ##\phi_{t}## which maps points ##x## in ##M^{n}## to points ##\phi_{t}(x)## in ##M^{n}## along the orbit of ##x## parameterised by time ##t##, we can define the differential ##\phi_{t*}## that maps the tangent vector of the vector field ##Y## at ##x## to some tangent vector at the point ##\phi_{t}x##. Now, there is only one tangent vector of the vector field ##\bf{Y}## at the point ##\phi_{t}x##, and this tangent vector is the vector ##{\bf{Y}}_{\phi_{t}x}##. This seems to suggest that ##{\bf{Y}}_{\phi_{t}x}## and ##\phi_{t*}{\bf{Y}}_{x}##.

    What am I missing?
     
  2. jcsd
  3. Oct 27, 2016 #2

    Ben Niehoff

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    Not sure what you mean by the phrase "tangent vector of the vector field". But anyway:

    ##Y_{\phi_t x}## is the vector that lives at ##\phi_t x##, whereas ##\phi_{t*} Y_x## is the vector that lives at ##x##, pushed forward by the flow ##\phi## to the point ##\phi_t x##.

    If you imagine vectors as tiny arrows living on your manifold, then the pushforward ##\phi_{t*}## acts by pushing both the head and the tail of the little arrow along the flow lines. So, the little arrow ##\phi_{t*} Y_x## is completely defined in terms of the little arrow ##Y_x## that lives at ##x##, given the flow ##\phi_t##.

    In contrast, the little arrow ##Y_{\phi_t x}## is the one that was already sitting at ##\phi_t x## to begin with, and hasn't been pushed along the flow.
     
  4. Oct 27, 2016 #3

    Orodruin

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    Just to put mathematics on what Ben said: ##\phi_t## defines a function from the manifold to itself. Any such function defines a map from the tangent space at ##x## to the tangent space at ##\phi_t(x)##. Taking ##X \in T_x M##, ##\phi_{t*}X## is defined by ##\phi_{t*}X[f] = X[f \circ \phi_t]## (note that ##f(\phi_t(x))## is a function on the manifold as long as ##f## is).
     
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