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Math Help - Infinitesimal generator

  1. #1
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    Infinitesimal generator

    Consider a one-parameter Lie group of transformations \mathbf{x}^* = \mathbf{X}(\mathbf{x},\varepsilon). The infinitesimals are given by \vec{\xi} = \left.\frac{\partial \mathbf{X}}{\partial \varepsilon}\right|_{\varepsilon=0} and the infinitesimal generator: X = \vec{\xi}\cdot\nabla. Let \mathbf{x}^* = (x^*,y^*). Now what if we have x^*=\frac{1}{\varepsilon}x and y^* = \varepsilon y? If we want to get the first component of \vec{\xi}, we need to differentiate x^* with respect to \varepsilon and evaluate it at \varepsilon=0 and we should divide by zero. How can it be solved?

    Thanks,
    Zoli
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  2. #2
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    Re: Infinitesimal generator

    This is not a Lie group. You need the identity meaning when $\varepsilon = 0$ then $x^*=x$. However, $\varepsilon = 1$ does work but you can re-parameterize by let $\varepsilon \rightarrow e^\varepsilon$ so that you can use $\varepsilon = 0$.
    Last edited by Jester; March 22nd 2014 at 04:16 AM.
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  3. #3
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    Re: Infinitesimal generator

    I should have thought of the identity transformation. Thank you.
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