Results 1 to 3 of 3

Thread: Infinitesimal generator

  1. #1
    Newbie
    Joined
    Feb 2013
    From
    Hungary
    Posts
    24

    Infinitesimal generator

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

    Thanks,
    Zoli
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,470
    Thanks
    83

    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; Mar 22nd 2014 at 04:16 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Feb 2013
    From
    Hungary
    Posts
    24

    Re: Infinitesimal generator

    I should have thought of the identity transformation. Thank you.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 6
    Last Post: Nov 1st 2012, 02:29 PM
  2. infinitesimal amount
    Posted in the New Users Forum
    Replies: 3
    Last Post: Apr 24th 2012, 12:29 PM
  3. Infinite set mapping to infinitesimal interval
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: Aug 1st 2010, 12:17 AM
  4. Infinitesimal 2. Second question!
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Mar 20th 2010, 12:05 AM
  5. Infinitesimal 2. First question! (Hard one...)
    Posted in the Calculus Forum
    Replies: 0
    Last Post: Mar 19th 2010, 07:08 AM

Search Tags


/mathhelpforum @mathhelpforum