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Hallo Math StackExchange users,

I have a little question concerning the exponential map and how one can apply this to infinitesimal transformations. I am an engineer in 2nd year and I want to write my bachelor thesis in fluid mechanics. I am using the book Applications of Lie Groups to Differential Equations (Peter J. Olver). As recommended by the author I started to read from Chapter 2.2 and was able to follow.

But now I am stuck at something. It's about the heat equation; I could follow the text until the derivation of the generators of symmetry.

$$ v_1 = \partial_x, \quad v_2 = \partial_t, \quad v_3 = u\partial_u, \quad v_4 = x\partial_x+2t\partial_t, \quad v_5 = 2t\partial_x-xu\partial_u, \quad \ldots $$

After this passage the one Parameter Groups $G_i$ which are generated by the $v_i$ are calculated by using $(x',t',u')=\exp(\epsilon v_i) (x,t,u)$. It is concluded that $G_1: (x+\epsilon,t,u)$, $G_2: (x,t+\epsilon,u)$, $G_3: (x,t,e^\epsilon u)$, $G_4: (e^\epsilon x, e^{2\epsilon}t,u)$, ... . Can someone explain how I can actually calculate these groups?

I tried this interpretation: $\exp(\epsilon v_i)(x,t,u)=(1+\epsilon v_i)(x,t,u) =(x+\epsilon v_i \cdot x,t+\epsilon v_i \cdot t, u \epsilon v_i \cdot u)$. I could recover $G_1$ and $G_2$ using this approach, while interpreting the operation between the exponential and the $(x,t,u)$ as a simple dot product. But from there I have no clue how to show that for $G_3$ and the other $G_i$.

I tried this for $v_3 = u\partial_u$. $\exp(\epsilon u\partial_u)(x,t,u)=(1+\epsilon u\partial_u+\frac{(\epsilon u\partial_u)^2}{2!}+...)(x,t,u)=(x,t,u+u\epsilon_u u)=(x,t,u+\epsilon u)$. I left out the higher order terms as I thought, that $\partial_u \partial_u u = 0$. If I had kept them in the series I could have recovered $G_3: (x,t,e^\epsilon u)$, but I could not explain it to myself why this should make any sense. But the textbook says $G_3: (x,t,e^\epsilon u)$. Could someone explain me (step by step) how one can get these groups? The book is really doing a bad job at explaining these steps. There is no explanation how to use this exponential map algorithmically.

Thank you for taking your time and reading my post :). I would be very thankful for any answer :D.

Travis Willse
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MrYouMath
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1 Answers1

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We simply want to compute the flow of each vector field $v_a$: For each $a$ and initial value $(x_0, t_0, u_0) \in \Bbb R^3_{xtu}$, consider the path $\gamma$ with $$\gamma(0) = (x_0, t_0, u_0) \quad \text{and} \quad \gamma'(s) = \epsilon v_a|_{\gamma(s)}.$$

For $a = 3$, for example, we have $$\epsilon v_3 = \epsilon u \partial_u ,$$ and so the above equation for $\gamma'$ is $$\left\{\begin{array}{rcl}x'(s) &=& 0 \\ t'(s) &=& 0 \\ u'(s) &=& \epsilon u \\\end{array}\right.$$ Solving this system is straightforward, and the action of the one parameter subgroup is: $$s \mapsto \gamma(s) = (x_0, t_0, u_0 e^{\epsilon s}).$$ In particular, we have the claimed action: $$\exp (\epsilon v_3) \cdot (x_0, t_0, u_0) = (x_0, t_0, u_0 e^{\epsilon}).$$

Travis Willse
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    Thank you alot for your response. But could you please explain how you get the ODEs? Or to be more pricise how would the ODEs look like if $v_a=f(x,t,u)\cdot \partial_x+g(x,t,u) \cdot \partial_t+h(x,t,u) \cdot \partial_u$? And why did the $u_0 e^{\epsilon s}$ become $u_0 e^\epsilon$ in the last line? – MrYouMath Sep 07 '15 at 19:32
  • In solving for the flow, we're looking for curves $\gamma$ whose tangent vector fields $\gamma'$ coincide with the value of the given vector field $\epsilon v_a$ at each point. If we write $\gamma(s)$ in components as $(F(s), G(s), H(s))$, then its tangent vector field is $F' \partial_x + G' \partial_t + H' \partial_u$. For the above condition to be satisfied, $\gamma(s)$ must satisfy the ODE system given by $F'(s) = \epsilon f(x(s), t(s), u(s))$ and the analogous equations for the other two components. – Travis Willse Sep 08 '15 at 00:38
  • Do you set $s=1$ in the last line? Why can you do this? Btw thank you for your explanations :). – MrYouMath Sep 08 '15 at 08:13
  • You're welcome. Evaluating at time $1$ is, if you like, the definition of the exponential map $\exp: \mathfrak{g} \to G$. – Travis Willse Sep 08 '15 at 15:47
  • If you have time, could you maybe have a look at my question concerning Lie Series? http://math.stackexchange.com/questions/1432104/how-to-properly-apply-the-lie-series – MrYouMath Sep 16 '15 at 18:47