Computing a derivative through Lie series












3












$begingroup$



Consider the $N$-dimensional autonomous system of ODEs
$$dot{x}= f(x),$$
where a locally unique solution $x(t)$, starting from the initial condition $x$, is denoted as $x(t)=phi(t,x)$. Assume that



$$Big(frac{partial}{partial{x}}phi(t,x)Big)f(x)=f(phi(t,x))$$



For the system above, assume that $f(x)$ is analytic (that is, its Taylor series converges to $f$ itself). Let the differential operator $L[xi]$ be defined as



$$L[xi]=f(x)boldsymbol{cdot}nabla{xi}=sum_{n=1}^{N}f_i(x)frac{partial{xi}}{partial{x_i}}$$



Show that $phi(t,x)$ can be expressed as



$$phi(t,x)=sum_{n=0}^{infty}frac{t^n}{n!}L^n[x]$$



where $L^n[xi]$ is the shorthand notation for



$$L^n[xi]=underbrace{L[L[cdots{L}[xi]}_{ntext{-times}}cdots]]$$




Potentially related questions:





  • How to properly apply the Lie Series

  • Exponential of a function times derivative

  • How to derive these Lie Series formulas


I'm stuck on how to approach this problem. Here is all the information that I have gathered so far -



Through this question, the one dimensional situation states that $e^{apartial}f(x)=f(a+x)$ (we can think of this as a shift operator).



Inside Ordinary Differential Equations and Dynamical Systems by Teschl, we have the following Lemma (Lemma $6.2$ on page $190$ of the text).



Lemma (Straightening out of vector fields): Suppose $f(x_0)neq0$. Then, there is a local coordinate transform $y=varphi(x)$ such that $dot{x}=f(x)$ is transformed to



$$dot{y}=(1,0,...,0)$$



Teschl list a similar problem on page $191$ (problem $6.5$ for one-parameter lie groups) in which he states that



Hint: The Taylor coefficients are the derivatives which can be obtained by
differentiating the differential equation.



So, I think that I need to apply what was done in this question alongside Lemma 6.2. I will have to consider what a vector field means in this context. I might be able to make the assumption that a vector field is just a linear operator. We are given that





  1. $dot{x}= f(x)$ is an autonomous system of ODEs

  2. $x(t)=phi(t,x)$

  3. $Big(frac{partial}{partial{x}}phi(t,x)Big)f(x)=f(phi(t,x))$

  4. $L[xi]=f(x)boldsymbol{cdot}nabla{xi}=sum_{n=1}^{N}f_i(x)frac{partial{xi}}{partial{x_i}}$


and we need to show that



$$phi(t,x)=sum_{n=0}^{infty}frac{t^n}{n!}L^n[x]$$



I also see that Roger Howe wrote a good introduction to lie theory in these notes (he goes through one-parameter lie groups on pages $604-606$).



This appears to be an extremely difficult problem for someone unfamiliar with lie theory. I am going to see if I can figure out a more direct approach.










share|cite|improve this question











$endgroup$

















    3












    $begingroup$



    Consider the $N$-dimensional autonomous system of ODEs
    $$dot{x}= f(x),$$
    where a locally unique solution $x(t)$, starting from the initial condition $x$, is denoted as $x(t)=phi(t,x)$. Assume that



    $$Big(frac{partial}{partial{x}}phi(t,x)Big)f(x)=f(phi(t,x))$$



    For the system above, assume that $f(x)$ is analytic (that is, its Taylor series converges to $f$ itself). Let the differential operator $L[xi]$ be defined as



    $$L[xi]=f(x)boldsymbol{cdot}nabla{xi}=sum_{n=1}^{N}f_i(x)frac{partial{xi}}{partial{x_i}}$$



    Show that $phi(t,x)$ can be expressed as



    $$phi(t,x)=sum_{n=0}^{infty}frac{t^n}{n!}L^n[x]$$



    where $L^n[xi]$ is the shorthand notation for



    $$L^n[xi]=underbrace{L[L[cdots{L}[xi]}_{ntext{-times}}cdots]]$$




    Potentially related questions:





    • How to properly apply the Lie Series

    • Exponential of a function times derivative

    • How to derive these Lie Series formulas


    I'm stuck on how to approach this problem. Here is all the information that I have gathered so far -



    Through this question, the one dimensional situation states that $e^{apartial}f(x)=f(a+x)$ (we can think of this as a shift operator).



    Inside Ordinary Differential Equations and Dynamical Systems by Teschl, we have the following Lemma (Lemma $6.2$ on page $190$ of the text).



    Lemma (Straightening out of vector fields): Suppose $f(x_0)neq0$. Then, there is a local coordinate transform $y=varphi(x)$ such that $dot{x}=f(x)$ is transformed to



    $$dot{y}=(1,0,...,0)$$



    Teschl list a similar problem on page $191$ (problem $6.5$ for one-parameter lie groups) in which he states that



    Hint: The Taylor coefficients are the derivatives which can be obtained by
    differentiating the differential equation.



    So, I think that I need to apply what was done in this question alongside Lemma 6.2. I will have to consider what a vector field means in this context. I might be able to make the assumption that a vector field is just a linear operator. We are given that





    1. $dot{x}= f(x)$ is an autonomous system of ODEs

    2. $x(t)=phi(t,x)$

    3. $Big(frac{partial}{partial{x}}phi(t,x)Big)f(x)=f(phi(t,x))$

    4. $L[xi]=f(x)boldsymbol{cdot}nabla{xi}=sum_{n=1}^{N}f_i(x)frac{partial{xi}}{partial{x_i}}$


    and we need to show that



    $$phi(t,x)=sum_{n=0}^{infty}frac{t^n}{n!}L^n[x]$$



    I also see that Roger Howe wrote a good introduction to lie theory in these notes (he goes through one-parameter lie groups on pages $604-606$).



    This appears to be an extremely difficult problem for someone unfamiliar with lie theory. I am going to see if I can figure out a more direct approach.










    share|cite|improve this question











    $endgroup$















      3












      3








      3





      $begingroup$



      Consider the $N$-dimensional autonomous system of ODEs
      $$dot{x}= f(x),$$
      where a locally unique solution $x(t)$, starting from the initial condition $x$, is denoted as $x(t)=phi(t,x)$. Assume that



      $$Big(frac{partial}{partial{x}}phi(t,x)Big)f(x)=f(phi(t,x))$$



      For the system above, assume that $f(x)$ is analytic (that is, its Taylor series converges to $f$ itself). Let the differential operator $L[xi]$ be defined as



      $$L[xi]=f(x)boldsymbol{cdot}nabla{xi}=sum_{n=1}^{N}f_i(x)frac{partial{xi}}{partial{x_i}}$$



      Show that $phi(t,x)$ can be expressed as



      $$phi(t,x)=sum_{n=0}^{infty}frac{t^n}{n!}L^n[x]$$



      where $L^n[xi]$ is the shorthand notation for



      $$L^n[xi]=underbrace{L[L[cdots{L}[xi]}_{ntext{-times}}cdots]]$$




      Potentially related questions:





      • How to properly apply the Lie Series

      • Exponential of a function times derivative

      • How to derive these Lie Series formulas


      I'm stuck on how to approach this problem. Here is all the information that I have gathered so far -



      Through this question, the one dimensional situation states that $e^{apartial}f(x)=f(a+x)$ (we can think of this as a shift operator).



      Inside Ordinary Differential Equations and Dynamical Systems by Teschl, we have the following Lemma (Lemma $6.2$ on page $190$ of the text).



      Lemma (Straightening out of vector fields): Suppose $f(x_0)neq0$. Then, there is a local coordinate transform $y=varphi(x)$ such that $dot{x}=f(x)$ is transformed to



      $$dot{y}=(1,0,...,0)$$



      Teschl list a similar problem on page $191$ (problem $6.5$ for one-parameter lie groups) in which he states that



      Hint: The Taylor coefficients are the derivatives which can be obtained by
      differentiating the differential equation.



      So, I think that I need to apply what was done in this question alongside Lemma 6.2. I will have to consider what a vector field means in this context. I might be able to make the assumption that a vector field is just a linear operator. We are given that





      1. $dot{x}= f(x)$ is an autonomous system of ODEs

      2. $x(t)=phi(t,x)$

      3. $Big(frac{partial}{partial{x}}phi(t,x)Big)f(x)=f(phi(t,x))$

      4. $L[xi]=f(x)boldsymbol{cdot}nabla{xi}=sum_{n=1}^{N}f_i(x)frac{partial{xi}}{partial{x_i}}$


      and we need to show that



      $$phi(t,x)=sum_{n=0}^{infty}frac{t^n}{n!}L^n[x]$$



      I also see that Roger Howe wrote a good introduction to lie theory in these notes (he goes through one-parameter lie groups on pages $604-606$).



      This appears to be an extremely difficult problem for someone unfamiliar with lie theory. I am going to see if I can figure out a more direct approach.










      share|cite|improve this question











      $endgroup$





      Consider the $N$-dimensional autonomous system of ODEs
      $$dot{x}= f(x),$$
      where a locally unique solution $x(t)$, starting from the initial condition $x$, is denoted as $x(t)=phi(t,x)$. Assume that



      $$Big(frac{partial}{partial{x}}phi(t,x)Big)f(x)=f(phi(t,x))$$



      For the system above, assume that $f(x)$ is analytic (that is, its Taylor series converges to $f$ itself). Let the differential operator $L[xi]$ be defined as



      $$L[xi]=f(x)boldsymbol{cdot}nabla{xi}=sum_{n=1}^{N}f_i(x)frac{partial{xi}}{partial{x_i}}$$



      Show that $phi(t,x)$ can be expressed as



      $$phi(t,x)=sum_{n=0}^{infty}frac{t^n}{n!}L^n[x]$$



      where $L^n[xi]$ is the shorthand notation for



      $$L^n[xi]=underbrace{L[L[cdots{L}[xi]}_{ntext{-times}}cdots]]$$




      Potentially related questions:





      • How to properly apply the Lie Series

      • Exponential of a function times derivative

      • How to derive these Lie Series formulas


      I'm stuck on how to approach this problem. Here is all the information that I have gathered so far -



      Through this question, the one dimensional situation states that $e^{apartial}f(x)=f(a+x)$ (we can think of this as a shift operator).



      Inside Ordinary Differential Equations and Dynamical Systems by Teschl, we have the following Lemma (Lemma $6.2$ on page $190$ of the text).



      Lemma (Straightening out of vector fields): Suppose $f(x_0)neq0$. Then, there is a local coordinate transform $y=varphi(x)$ such that $dot{x}=f(x)$ is transformed to



      $$dot{y}=(1,0,...,0)$$



      Teschl list a similar problem on page $191$ (problem $6.5$ for one-parameter lie groups) in which he states that



      Hint: The Taylor coefficients are the derivatives which can be obtained by
      differentiating the differential equation.



      So, I think that I need to apply what was done in this question alongside Lemma 6.2. I will have to consider what a vector field means in this context. I might be able to make the assumption that a vector field is just a linear operator. We are given that





      1. $dot{x}= f(x)$ is an autonomous system of ODEs

      2. $x(t)=phi(t,x)$

      3. $Big(frac{partial}{partial{x}}phi(t,x)Big)f(x)=f(phi(t,x))$

      4. $L[xi]=f(x)boldsymbol{cdot}nabla{xi}=sum_{n=1}^{N}f_i(x)frac{partial{xi}}{partial{x_i}}$


      and we need to show that



      $$phi(t,x)=sum_{n=0}^{infty}frac{t^n}{n!}L^n[x]$$



      I also see that Roger Howe wrote a good introduction to lie theory in these notes (he goes through one-parameter lie groups on pages $604-606$).



      This appears to be an extremely difficult problem for someone unfamiliar with lie theory. I am going to see if I can figure out a more direct approach.







      ordinary-differential-equations lie-groups differential-operators






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 3 at 20:05









      Bernard

      124k741118




      124k741118










      asked Jan 3 at 19:34









      Axion004Axion004

      405413




      405413






















          2 Answers
          2






          active

          oldest

          votes


















          1












          $begingroup$

          For any differentiable function $B:Bbb R^ntoBbb R^n$ we know from chain rule and differential equation that
          begin{align}
          frac{∂}{∂t}B(ϕ(t,x))&=frac{∂B}{∂x}(ϕ(t,x))cdot frac{∂}{∂t}ϕ(t,x)
          \
          &=frac{∂B}{∂x}(ϕ(t,x))cdot f(ϕ(t,x))
          \
          &=sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))=L_{ϕ(t,x)}[B].
          end{align}

          So along a solution we get $frac{∂}{∂t}=L_{ϕ(t,x)}$. Now apply this to the translation operator resp. the Taylor expansion
          $$
          ϕ(t,x)=expleft(tfrac{∂}{∂s}right)ϕ(s,x)Big|_{s=0}
          =expleft(tL_{ϕ(s,x)}right)[ϕ(s,x)]Big|_{s=0}
          =expleft(tL_{x}right)[x]Big|_{s=0}
          $$

          The same remains true if you replace the exponential by the exponential series.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I'm having trouble following $frac{∂B}{∂x}(ϕ(t,x))cdot f(ϕ(t,x))=sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))=L_{ϕ(t,x)}[B]$. For the first equality, $frac{∂B}{∂x}(ϕ(t,x))cdot f(ϕ(t,x))=sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))$, you must be taking a partial derivative in every direction (since we are working in $Bbb R^n$). I don't see how one can justify $sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))=L_{ϕ(t,x)}[B]$. This must follow from the definition of the differential operator.
            $endgroup$
            – Axion004
            Jan 4 at 18:02












          • $begingroup$
            Simple manipulation of linear Taylor polynomials gives $$B(x(t+h))=B(x(t)+f(x(t))h+O(h^2)=B(x(t))+B'(x(t))f(x(t))h+O(h^2).$$ Inside $B'(x)v$ with $v=f(x)h$ is the directional derivative in direction $v$, $B'(x)v=sum v_ifrac∂{∂x_i}B(x)$. Replacing back $v=f(x)h$ gives exactly the definition of $L$, so that $B(x(t+h))=B(x(t))+L[B](x(t))h+O(h^2)$. There is nothing more to it.
            $endgroup$
            – LutzL
            Jan 4 at 18:14










          • $begingroup$
            Perhaps, to avoid using the Taylor series expansion, one could apply the definition of the directional derivative and conclude that $dfrac{partialxi(x)}{partial{x}}cdot{f(x)}=nabla{xi(x)}boldsymbol{cdot}f(x)=sum_{n=1}^{N}frac{partial{xi}}{partial{x_i}}f_i(x)$.
            $endgroup$
            – Axion004
            Jan 6 at 1:05










          • $begingroup$
            I'm guessing that there is a logical reason why $phi(t,x)=expleft(tfrac{partial}{partial{s}}right)phi(s,x)Big|_{s=0}$. I tried reviewing it tonight and couldn't see how this was formed. I coudn't derive this starting from $frac{partial}{partial{x}}phi(t,x)f(x)=f(phi(t,x))$.
            $endgroup$
            – Axion004
            Jan 6 at 2:40












          • $begingroup$
            This is just the simple Taylor expansion $x(t)=sumfrac{x^{(k)}}{k!}t^k=(exp(tD)x)(0)$. $t$ is here a constant, so it looks bad to have the derivative for $t$, thus changing it to $s$.
            $endgroup$
            – LutzL
            Jan 6 at 8:56



















          0












          $begingroup$

          Here is my interpretation of the first answer.



          Suppose we have a differentiable function $xi:Bbb R^ntoBbb R^n$ where $frac{partial}{partial{t}}phi(t,x)=f(phi(t,x))$. We know by the chain rule that
          begin{align*}
          frac{partial}{partial{t}}xi(phi(t,x))&=frac{partialxi}{partial{x}}(phi(t,x))cdot frac{partial}{partial{t}}phi(t,x)
          \
          &=frac{partialxi}{partial{x}}(phi(t,x))cdot f(phi(t,x))
          \
          &=frac{partialxi}{partial{x}}(x(t))cdot f(x(t))
          end{align*}



          where $dfrac{partialxi(x)}{partial{x}}cdot{f(x)}$ is the directional derivative of the function $xi$ in the direction of f. This is defined as



          $$dfrac{partialxi(x)}{partial{x}}cdot{f(x)}=nabla{xi(x)}boldsymbol{cdot}f(x)=sum_{n=1}^{N}frac{partial{xi}}{partial{x_i}}f_i(x)$$



          Therefore,



          begin{align*}
          frac{partial}{partial{t}}xi(phi(t,x))&=frac{partialxi}{partial{x}}(x(t))cdot f(x(t))\
          &=sum_{i=1}^n frac{partialxi}{partial{x_i}}(x(t)) f_i(x(t))
          \&=sum_{i=1}^n f_i(phi(t,x))frac{partialxi}{partial{x_i}}(phi(t,x))
          =L_{phi(t,x)}[xi]
          end{align*}



          So along a solution we get $frac{partial}{partial{t}}=L_{phi(t,x)}$. Next, observe that we can write $x(a)$ in series notation as



          $$x(a)=sum_{n=0}^inftyfrac{x^{(n)}}{n!}a^n$$



          We also know that



          $$exp(a)=sum_{n=0}^inftyfrac{a^n}{n!}$$



          Applying $exp(a)$ to the differential operator $afrac{partial}{partial{t}}$ produces



          $$expBig(afrac{partial}{partial{t}}Big)=sum_{n=0}^inftyfrac{a^n}{n!}Big(frac{partial}{partial{t}}Big)^n$$



          Hence if we evaluate this at $x(t)$ we have



          $$expBig(afrac{partial}{partial{t}}Big)x(t)=sum_{n=0}^inftyfrac{a^n}{n!}Big(frac{partial{x(t)}}{partial{t}}Big)^n =sum_{n=0}^inftyfrac{a^n}{n!}x^{n}(t)=sum_{n=0}^inftyfrac{x^{n}(t)}{n!}a^n$$



          Where it is necessary for $t=0$ in order to compute the Maclaurin expansion. Therefore,



          $$expBig(afrac{partial}{partial{t}}Big)x(t)Big|_{t=0}=sum_{n=0}^inftyfrac{x^{n}(t)}{n!}a^n=x(a)$$



          Noting that $x(t)=phi(t,x)$ and changing the constants $arightarrow{t}$, $trightarrow{s}$ so that $x(a)=x(t)$ and $expBig(afrac{partial}{partial{t}}Big)x(t) = expBig(tfrac{partial}{partial{s}}Big)x(s)$ allows us to write



          $$
          phi(t,x)=expleft(tfrac{partial}{partial{s}}right)phi(s,x)Big|_{s=0}
          =expleft(tL_{phi(s,x)}right)[phi(s,x)]Big|_{s=0}
          =expleft(tL_{x}right)[x]
          $$

          If we then replace the exponential with the exponential series, we have
          $$
          phi(t,x)=expleft(tL_{x}right)[x]=Big(sum_{n = 0}^{infty} frac{left(tL_{x}right)^n}{n!}Big)[x]=sum_{n=0}^{infty}frac{t^n}{n!}L^n[x]$$






          share|cite|improve this answer











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            2 Answers
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            2 Answers
            2






            active

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            active

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            active

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            1












            $begingroup$

            For any differentiable function $B:Bbb R^ntoBbb R^n$ we know from chain rule and differential equation that
            begin{align}
            frac{∂}{∂t}B(ϕ(t,x))&=frac{∂B}{∂x}(ϕ(t,x))cdot frac{∂}{∂t}ϕ(t,x)
            \
            &=frac{∂B}{∂x}(ϕ(t,x))cdot f(ϕ(t,x))
            \
            &=sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))=L_{ϕ(t,x)}[B].
            end{align}

            So along a solution we get $frac{∂}{∂t}=L_{ϕ(t,x)}$. Now apply this to the translation operator resp. the Taylor expansion
            $$
            ϕ(t,x)=expleft(tfrac{∂}{∂s}right)ϕ(s,x)Big|_{s=0}
            =expleft(tL_{ϕ(s,x)}right)[ϕ(s,x)]Big|_{s=0}
            =expleft(tL_{x}right)[x]Big|_{s=0}
            $$

            The same remains true if you replace the exponential by the exponential series.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              I'm having trouble following $frac{∂B}{∂x}(ϕ(t,x))cdot f(ϕ(t,x))=sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))=L_{ϕ(t,x)}[B]$. For the first equality, $frac{∂B}{∂x}(ϕ(t,x))cdot f(ϕ(t,x))=sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))$, you must be taking a partial derivative in every direction (since we are working in $Bbb R^n$). I don't see how one can justify $sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))=L_{ϕ(t,x)}[B]$. This must follow from the definition of the differential operator.
              $endgroup$
              – Axion004
              Jan 4 at 18:02












            • $begingroup$
              Simple manipulation of linear Taylor polynomials gives $$B(x(t+h))=B(x(t)+f(x(t))h+O(h^2)=B(x(t))+B'(x(t))f(x(t))h+O(h^2).$$ Inside $B'(x)v$ with $v=f(x)h$ is the directional derivative in direction $v$, $B'(x)v=sum v_ifrac∂{∂x_i}B(x)$. Replacing back $v=f(x)h$ gives exactly the definition of $L$, so that $B(x(t+h))=B(x(t))+L[B](x(t))h+O(h^2)$. There is nothing more to it.
              $endgroup$
              – LutzL
              Jan 4 at 18:14










            • $begingroup$
              Perhaps, to avoid using the Taylor series expansion, one could apply the definition of the directional derivative and conclude that $dfrac{partialxi(x)}{partial{x}}cdot{f(x)}=nabla{xi(x)}boldsymbol{cdot}f(x)=sum_{n=1}^{N}frac{partial{xi}}{partial{x_i}}f_i(x)$.
              $endgroup$
              – Axion004
              Jan 6 at 1:05










            • $begingroup$
              I'm guessing that there is a logical reason why $phi(t,x)=expleft(tfrac{partial}{partial{s}}right)phi(s,x)Big|_{s=0}$. I tried reviewing it tonight and couldn't see how this was formed. I coudn't derive this starting from $frac{partial}{partial{x}}phi(t,x)f(x)=f(phi(t,x))$.
              $endgroup$
              – Axion004
              Jan 6 at 2:40












            • $begingroup$
              This is just the simple Taylor expansion $x(t)=sumfrac{x^{(k)}}{k!}t^k=(exp(tD)x)(0)$. $t$ is here a constant, so it looks bad to have the derivative for $t$, thus changing it to $s$.
              $endgroup$
              – LutzL
              Jan 6 at 8:56
















            1












            $begingroup$

            For any differentiable function $B:Bbb R^ntoBbb R^n$ we know from chain rule and differential equation that
            begin{align}
            frac{∂}{∂t}B(ϕ(t,x))&=frac{∂B}{∂x}(ϕ(t,x))cdot frac{∂}{∂t}ϕ(t,x)
            \
            &=frac{∂B}{∂x}(ϕ(t,x))cdot f(ϕ(t,x))
            \
            &=sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))=L_{ϕ(t,x)}[B].
            end{align}

            So along a solution we get $frac{∂}{∂t}=L_{ϕ(t,x)}$. Now apply this to the translation operator resp. the Taylor expansion
            $$
            ϕ(t,x)=expleft(tfrac{∂}{∂s}right)ϕ(s,x)Big|_{s=0}
            =expleft(tL_{ϕ(s,x)}right)[ϕ(s,x)]Big|_{s=0}
            =expleft(tL_{x}right)[x]Big|_{s=0}
            $$

            The same remains true if you replace the exponential by the exponential series.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              I'm having trouble following $frac{∂B}{∂x}(ϕ(t,x))cdot f(ϕ(t,x))=sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))=L_{ϕ(t,x)}[B]$. For the first equality, $frac{∂B}{∂x}(ϕ(t,x))cdot f(ϕ(t,x))=sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))$, you must be taking a partial derivative in every direction (since we are working in $Bbb R^n$). I don't see how one can justify $sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))=L_{ϕ(t,x)}[B]$. This must follow from the definition of the differential operator.
              $endgroup$
              – Axion004
              Jan 4 at 18:02












            • $begingroup$
              Simple manipulation of linear Taylor polynomials gives $$B(x(t+h))=B(x(t)+f(x(t))h+O(h^2)=B(x(t))+B'(x(t))f(x(t))h+O(h^2).$$ Inside $B'(x)v$ with $v=f(x)h$ is the directional derivative in direction $v$, $B'(x)v=sum v_ifrac∂{∂x_i}B(x)$. Replacing back $v=f(x)h$ gives exactly the definition of $L$, so that $B(x(t+h))=B(x(t))+L[B](x(t))h+O(h^2)$. There is nothing more to it.
              $endgroup$
              – LutzL
              Jan 4 at 18:14










            • $begingroup$
              Perhaps, to avoid using the Taylor series expansion, one could apply the definition of the directional derivative and conclude that $dfrac{partialxi(x)}{partial{x}}cdot{f(x)}=nabla{xi(x)}boldsymbol{cdot}f(x)=sum_{n=1}^{N}frac{partial{xi}}{partial{x_i}}f_i(x)$.
              $endgroup$
              – Axion004
              Jan 6 at 1:05










            • $begingroup$
              I'm guessing that there is a logical reason why $phi(t,x)=expleft(tfrac{partial}{partial{s}}right)phi(s,x)Big|_{s=0}$. I tried reviewing it tonight and couldn't see how this was formed. I coudn't derive this starting from $frac{partial}{partial{x}}phi(t,x)f(x)=f(phi(t,x))$.
              $endgroup$
              – Axion004
              Jan 6 at 2:40












            • $begingroup$
              This is just the simple Taylor expansion $x(t)=sumfrac{x^{(k)}}{k!}t^k=(exp(tD)x)(0)$. $t$ is here a constant, so it looks bad to have the derivative for $t$, thus changing it to $s$.
              $endgroup$
              – LutzL
              Jan 6 at 8:56














            1












            1








            1





            $begingroup$

            For any differentiable function $B:Bbb R^ntoBbb R^n$ we know from chain rule and differential equation that
            begin{align}
            frac{∂}{∂t}B(ϕ(t,x))&=frac{∂B}{∂x}(ϕ(t,x))cdot frac{∂}{∂t}ϕ(t,x)
            \
            &=frac{∂B}{∂x}(ϕ(t,x))cdot f(ϕ(t,x))
            \
            &=sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))=L_{ϕ(t,x)}[B].
            end{align}

            So along a solution we get $frac{∂}{∂t}=L_{ϕ(t,x)}$. Now apply this to the translation operator resp. the Taylor expansion
            $$
            ϕ(t,x)=expleft(tfrac{∂}{∂s}right)ϕ(s,x)Big|_{s=0}
            =expleft(tL_{ϕ(s,x)}right)[ϕ(s,x)]Big|_{s=0}
            =expleft(tL_{x}right)[x]Big|_{s=0}
            $$

            The same remains true if you replace the exponential by the exponential series.






            share|cite|improve this answer









            $endgroup$



            For any differentiable function $B:Bbb R^ntoBbb R^n$ we know from chain rule and differential equation that
            begin{align}
            frac{∂}{∂t}B(ϕ(t,x))&=frac{∂B}{∂x}(ϕ(t,x))cdot frac{∂}{∂t}ϕ(t,x)
            \
            &=frac{∂B}{∂x}(ϕ(t,x))cdot f(ϕ(t,x))
            \
            &=sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))=L_{ϕ(t,x)}[B].
            end{align}

            So along a solution we get $frac{∂}{∂t}=L_{ϕ(t,x)}$. Now apply this to the translation operator resp. the Taylor expansion
            $$
            ϕ(t,x)=expleft(tfrac{∂}{∂s}right)ϕ(s,x)Big|_{s=0}
            =expleft(tL_{ϕ(s,x)}right)[ϕ(s,x)]Big|_{s=0}
            =expleft(tL_{x}right)[x]Big|_{s=0}
            $$

            The same remains true if you replace the exponential by the exponential series.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Jan 3 at 21:40









            LutzLLutzL

            60.1k42057




            60.1k42057












            • $begingroup$
              I'm having trouble following $frac{∂B}{∂x}(ϕ(t,x))cdot f(ϕ(t,x))=sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))=L_{ϕ(t,x)}[B]$. For the first equality, $frac{∂B}{∂x}(ϕ(t,x))cdot f(ϕ(t,x))=sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))$, you must be taking a partial derivative in every direction (since we are working in $Bbb R^n$). I don't see how one can justify $sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))=L_{ϕ(t,x)}[B]$. This must follow from the definition of the differential operator.
              $endgroup$
              – Axion004
              Jan 4 at 18:02












            • $begingroup$
              Simple manipulation of linear Taylor polynomials gives $$B(x(t+h))=B(x(t)+f(x(t))h+O(h^2)=B(x(t))+B'(x(t))f(x(t))h+O(h^2).$$ Inside $B'(x)v$ with $v=f(x)h$ is the directional derivative in direction $v$, $B'(x)v=sum v_ifrac∂{∂x_i}B(x)$. Replacing back $v=f(x)h$ gives exactly the definition of $L$, so that $B(x(t+h))=B(x(t))+L[B](x(t))h+O(h^2)$. There is nothing more to it.
              $endgroup$
              – LutzL
              Jan 4 at 18:14










            • $begingroup$
              Perhaps, to avoid using the Taylor series expansion, one could apply the definition of the directional derivative and conclude that $dfrac{partialxi(x)}{partial{x}}cdot{f(x)}=nabla{xi(x)}boldsymbol{cdot}f(x)=sum_{n=1}^{N}frac{partial{xi}}{partial{x_i}}f_i(x)$.
              $endgroup$
              – Axion004
              Jan 6 at 1:05










            • $begingroup$
              I'm guessing that there is a logical reason why $phi(t,x)=expleft(tfrac{partial}{partial{s}}right)phi(s,x)Big|_{s=0}$. I tried reviewing it tonight and couldn't see how this was formed. I coudn't derive this starting from $frac{partial}{partial{x}}phi(t,x)f(x)=f(phi(t,x))$.
              $endgroup$
              – Axion004
              Jan 6 at 2:40












            • $begingroup$
              This is just the simple Taylor expansion $x(t)=sumfrac{x^{(k)}}{k!}t^k=(exp(tD)x)(0)$. $t$ is here a constant, so it looks bad to have the derivative for $t$, thus changing it to $s$.
              $endgroup$
              – LutzL
              Jan 6 at 8:56


















            • $begingroup$
              I'm having trouble following $frac{∂B}{∂x}(ϕ(t,x))cdot f(ϕ(t,x))=sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))=L_{ϕ(t,x)}[B]$. For the first equality, $frac{∂B}{∂x}(ϕ(t,x))cdot f(ϕ(t,x))=sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))$, you must be taking a partial derivative in every direction (since we are working in $Bbb R^n$). I don't see how one can justify $sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))=L_{ϕ(t,x)}[B]$. This must follow from the definition of the differential operator.
              $endgroup$
              – Axion004
              Jan 4 at 18:02












            • $begingroup$
              Simple manipulation of linear Taylor polynomials gives $$B(x(t+h))=B(x(t)+f(x(t))h+O(h^2)=B(x(t))+B'(x(t))f(x(t))h+O(h^2).$$ Inside $B'(x)v$ with $v=f(x)h$ is the directional derivative in direction $v$, $B'(x)v=sum v_ifrac∂{∂x_i}B(x)$. Replacing back $v=f(x)h$ gives exactly the definition of $L$, so that $B(x(t+h))=B(x(t))+L[B](x(t))h+O(h^2)$. There is nothing more to it.
              $endgroup$
              – LutzL
              Jan 4 at 18:14










            • $begingroup$
              Perhaps, to avoid using the Taylor series expansion, one could apply the definition of the directional derivative and conclude that $dfrac{partialxi(x)}{partial{x}}cdot{f(x)}=nabla{xi(x)}boldsymbol{cdot}f(x)=sum_{n=1}^{N}frac{partial{xi}}{partial{x_i}}f_i(x)$.
              $endgroup$
              – Axion004
              Jan 6 at 1:05










            • $begingroup$
              I'm guessing that there is a logical reason why $phi(t,x)=expleft(tfrac{partial}{partial{s}}right)phi(s,x)Big|_{s=0}$. I tried reviewing it tonight and couldn't see how this was formed. I coudn't derive this starting from $frac{partial}{partial{x}}phi(t,x)f(x)=f(phi(t,x))$.
              $endgroup$
              – Axion004
              Jan 6 at 2:40












            • $begingroup$
              This is just the simple Taylor expansion $x(t)=sumfrac{x^{(k)}}{k!}t^k=(exp(tD)x)(0)$. $t$ is here a constant, so it looks bad to have the derivative for $t$, thus changing it to $s$.
              $endgroup$
              – LutzL
              Jan 6 at 8:56
















            $begingroup$
            I'm having trouble following $frac{∂B}{∂x}(ϕ(t,x))cdot f(ϕ(t,x))=sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))=L_{ϕ(t,x)}[B]$. For the first equality, $frac{∂B}{∂x}(ϕ(t,x))cdot f(ϕ(t,x))=sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))$, you must be taking a partial derivative in every direction (since we are working in $Bbb R^n$). I don't see how one can justify $sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))=L_{ϕ(t,x)}[B]$. This must follow from the definition of the differential operator.
            $endgroup$
            – Axion004
            Jan 4 at 18:02






            $begingroup$
            I'm having trouble following $frac{∂B}{∂x}(ϕ(t,x))cdot f(ϕ(t,x))=sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))=L_{ϕ(t,x)}[B]$. For the first equality, $frac{∂B}{∂x}(ϕ(t,x))cdot f(ϕ(t,x))=sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))$, you must be taking a partial derivative in every direction (since we are working in $Bbb R^n$). I don't see how one can justify $sum_{i=1}^n frac{∂B}{∂x_i}(ϕ(t,x)) f_i(ϕ(t,x))=L_{ϕ(t,x)}[B]$. This must follow from the definition of the differential operator.
            $endgroup$
            – Axion004
            Jan 4 at 18:02














            $begingroup$
            Simple manipulation of linear Taylor polynomials gives $$B(x(t+h))=B(x(t)+f(x(t))h+O(h^2)=B(x(t))+B'(x(t))f(x(t))h+O(h^2).$$ Inside $B'(x)v$ with $v=f(x)h$ is the directional derivative in direction $v$, $B'(x)v=sum v_ifrac∂{∂x_i}B(x)$. Replacing back $v=f(x)h$ gives exactly the definition of $L$, so that $B(x(t+h))=B(x(t))+L[B](x(t))h+O(h^2)$. There is nothing more to it.
            $endgroup$
            – LutzL
            Jan 4 at 18:14




            $begingroup$
            Simple manipulation of linear Taylor polynomials gives $$B(x(t+h))=B(x(t)+f(x(t))h+O(h^2)=B(x(t))+B'(x(t))f(x(t))h+O(h^2).$$ Inside $B'(x)v$ with $v=f(x)h$ is the directional derivative in direction $v$, $B'(x)v=sum v_ifrac∂{∂x_i}B(x)$. Replacing back $v=f(x)h$ gives exactly the definition of $L$, so that $B(x(t+h))=B(x(t))+L[B](x(t))h+O(h^2)$. There is nothing more to it.
            $endgroup$
            – LutzL
            Jan 4 at 18:14












            $begingroup$
            Perhaps, to avoid using the Taylor series expansion, one could apply the definition of the directional derivative and conclude that $dfrac{partialxi(x)}{partial{x}}cdot{f(x)}=nabla{xi(x)}boldsymbol{cdot}f(x)=sum_{n=1}^{N}frac{partial{xi}}{partial{x_i}}f_i(x)$.
            $endgroup$
            – Axion004
            Jan 6 at 1:05




            $begingroup$
            Perhaps, to avoid using the Taylor series expansion, one could apply the definition of the directional derivative and conclude that $dfrac{partialxi(x)}{partial{x}}cdot{f(x)}=nabla{xi(x)}boldsymbol{cdot}f(x)=sum_{n=1}^{N}frac{partial{xi}}{partial{x_i}}f_i(x)$.
            $endgroup$
            – Axion004
            Jan 6 at 1:05












            $begingroup$
            I'm guessing that there is a logical reason why $phi(t,x)=expleft(tfrac{partial}{partial{s}}right)phi(s,x)Big|_{s=0}$. I tried reviewing it tonight and couldn't see how this was formed. I coudn't derive this starting from $frac{partial}{partial{x}}phi(t,x)f(x)=f(phi(t,x))$.
            $endgroup$
            – Axion004
            Jan 6 at 2:40






            $begingroup$
            I'm guessing that there is a logical reason why $phi(t,x)=expleft(tfrac{partial}{partial{s}}right)phi(s,x)Big|_{s=0}$. I tried reviewing it tonight and couldn't see how this was formed. I coudn't derive this starting from $frac{partial}{partial{x}}phi(t,x)f(x)=f(phi(t,x))$.
            $endgroup$
            – Axion004
            Jan 6 at 2:40














            $begingroup$
            This is just the simple Taylor expansion $x(t)=sumfrac{x^{(k)}}{k!}t^k=(exp(tD)x)(0)$. $t$ is here a constant, so it looks bad to have the derivative for $t$, thus changing it to $s$.
            $endgroup$
            – LutzL
            Jan 6 at 8:56




            $begingroup$
            This is just the simple Taylor expansion $x(t)=sumfrac{x^{(k)}}{k!}t^k=(exp(tD)x)(0)$. $t$ is here a constant, so it looks bad to have the derivative for $t$, thus changing it to $s$.
            $endgroup$
            – LutzL
            Jan 6 at 8:56











            0












            $begingroup$

            Here is my interpretation of the first answer.



            Suppose we have a differentiable function $xi:Bbb R^ntoBbb R^n$ where $frac{partial}{partial{t}}phi(t,x)=f(phi(t,x))$. We know by the chain rule that
            begin{align*}
            frac{partial}{partial{t}}xi(phi(t,x))&=frac{partialxi}{partial{x}}(phi(t,x))cdot frac{partial}{partial{t}}phi(t,x)
            \
            &=frac{partialxi}{partial{x}}(phi(t,x))cdot f(phi(t,x))
            \
            &=frac{partialxi}{partial{x}}(x(t))cdot f(x(t))
            end{align*}



            where $dfrac{partialxi(x)}{partial{x}}cdot{f(x)}$ is the directional derivative of the function $xi$ in the direction of f. This is defined as



            $$dfrac{partialxi(x)}{partial{x}}cdot{f(x)}=nabla{xi(x)}boldsymbol{cdot}f(x)=sum_{n=1}^{N}frac{partial{xi}}{partial{x_i}}f_i(x)$$



            Therefore,



            begin{align*}
            frac{partial}{partial{t}}xi(phi(t,x))&=frac{partialxi}{partial{x}}(x(t))cdot f(x(t))\
            &=sum_{i=1}^n frac{partialxi}{partial{x_i}}(x(t)) f_i(x(t))
            \&=sum_{i=1}^n f_i(phi(t,x))frac{partialxi}{partial{x_i}}(phi(t,x))
            =L_{phi(t,x)}[xi]
            end{align*}



            So along a solution we get $frac{partial}{partial{t}}=L_{phi(t,x)}$. Next, observe that we can write $x(a)$ in series notation as



            $$x(a)=sum_{n=0}^inftyfrac{x^{(n)}}{n!}a^n$$



            We also know that



            $$exp(a)=sum_{n=0}^inftyfrac{a^n}{n!}$$



            Applying $exp(a)$ to the differential operator $afrac{partial}{partial{t}}$ produces



            $$expBig(afrac{partial}{partial{t}}Big)=sum_{n=0}^inftyfrac{a^n}{n!}Big(frac{partial}{partial{t}}Big)^n$$



            Hence if we evaluate this at $x(t)$ we have



            $$expBig(afrac{partial}{partial{t}}Big)x(t)=sum_{n=0}^inftyfrac{a^n}{n!}Big(frac{partial{x(t)}}{partial{t}}Big)^n =sum_{n=0}^inftyfrac{a^n}{n!}x^{n}(t)=sum_{n=0}^inftyfrac{x^{n}(t)}{n!}a^n$$



            Where it is necessary for $t=0$ in order to compute the Maclaurin expansion. Therefore,



            $$expBig(afrac{partial}{partial{t}}Big)x(t)Big|_{t=0}=sum_{n=0}^inftyfrac{x^{n}(t)}{n!}a^n=x(a)$$



            Noting that $x(t)=phi(t,x)$ and changing the constants $arightarrow{t}$, $trightarrow{s}$ so that $x(a)=x(t)$ and $expBig(afrac{partial}{partial{t}}Big)x(t) = expBig(tfrac{partial}{partial{s}}Big)x(s)$ allows us to write



            $$
            phi(t,x)=expleft(tfrac{partial}{partial{s}}right)phi(s,x)Big|_{s=0}
            =expleft(tL_{phi(s,x)}right)[phi(s,x)]Big|_{s=0}
            =expleft(tL_{x}right)[x]
            $$

            If we then replace the exponential with the exponential series, we have
            $$
            phi(t,x)=expleft(tL_{x}right)[x]=Big(sum_{n = 0}^{infty} frac{left(tL_{x}right)^n}{n!}Big)[x]=sum_{n=0}^{infty}frac{t^n}{n!}L^n[x]$$






            share|cite|improve this answer











            $endgroup$


















              0












              $begingroup$

              Here is my interpretation of the first answer.



              Suppose we have a differentiable function $xi:Bbb R^ntoBbb R^n$ where $frac{partial}{partial{t}}phi(t,x)=f(phi(t,x))$. We know by the chain rule that
              begin{align*}
              frac{partial}{partial{t}}xi(phi(t,x))&=frac{partialxi}{partial{x}}(phi(t,x))cdot frac{partial}{partial{t}}phi(t,x)
              \
              &=frac{partialxi}{partial{x}}(phi(t,x))cdot f(phi(t,x))
              \
              &=frac{partialxi}{partial{x}}(x(t))cdot f(x(t))
              end{align*}



              where $dfrac{partialxi(x)}{partial{x}}cdot{f(x)}$ is the directional derivative of the function $xi$ in the direction of f. This is defined as



              $$dfrac{partialxi(x)}{partial{x}}cdot{f(x)}=nabla{xi(x)}boldsymbol{cdot}f(x)=sum_{n=1}^{N}frac{partial{xi}}{partial{x_i}}f_i(x)$$



              Therefore,



              begin{align*}
              frac{partial}{partial{t}}xi(phi(t,x))&=frac{partialxi}{partial{x}}(x(t))cdot f(x(t))\
              &=sum_{i=1}^n frac{partialxi}{partial{x_i}}(x(t)) f_i(x(t))
              \&=sum_{i=1}^n f_i(phi(t,x))frac{partialxi}{partial{x_i}}(phi(t,x))
              =L_{phi(t,x)}[xi]
              end{align*}



              So along a solution we get $frac{partial}{partial{t}}=L_{phi(t,x)}$. Next, observe that we can write $x(a)$ in series notation as



              $$x(a)=sum_{n=0}^inftyfrac{x^{(n)}}{n!}a^n$$



              We also know that



              $$exp(a)=sum_{n=0}^inftyfrac{a^n}{n!}$$



              Applying $exp(a)$ to the differential operator $afrac{partial}{partial{t}}$ produces



              $$expBig(afrac{partial}{partial{t}}Big)=sum_{n=0}^inftyfrac{a^n}{n!}Big(frac{partial}{partial{t}}Big)^n$$



              Hence if we evaluate this at $x(t)$ we have



              $$expBig(afrac{partial}{partial{t}}Big)x(t)=sum_{n=0}^inftyfrac{a^n}{n!}Big(frac{partial{x(t)}}{partial{t}}Big)^n =sum_{n=0}^inftyfrac{a^n}{n!}x^{n}(t)=sum_{n=0}^inftyfrac{x^{n}(t)}{n!}a^n$$



              Where it is necessary for $t=0$ in order to compute the Maclaurin expansion. Therefore,



              $$expBig(afrac{partial}{partial{t}}Big)x(t)Big|_{t=0}=sum_{n=0}^inftyfrac{x^{n}(t)}{n!}a^n=x(a)$$



              Noting that $x(t)=phi(t,x)$ and changing the constants $arightarrow{t}$, $trightarrow{s}$ so that $x(a)=x(t)$ and $expBig(afrac{partial}{partial{t}}Big)x(t) = expBig(tfrac{partial}{partial{s}}Big)x(s)$ allows us to write



              $$
              phi(t,x)=expleft(tfrac{partial}{partial{s}}right)phi(s,x)Big|_{s=0}
              =expleft(tL_{phi(s,x)}right)[phi(s,x)]Big|_{s=0}
              =expleft(tL_{x}right)[x]
              $$

              If we then replace the exponential with the exponential series, we have
              $$
              phi(t,x)=expleft(tL_{x}right)[x]=Big(sum_{n = 0}^{infty} frac{left(tL_{x}right)^n}{n!}Big)[x]=sum_{n=0}^{infty}frac{t^n}{n!}L^n[x]$$






              share|cite|improve this answer











              $endgroup$
















                0












                0








                0





                $begingroup$

                Here is my interpretation of the first answer.



                Suppose we have a differentiable function $xi:Bbb R^ntoBbb R^n$ where $frac{partial}{partial{t}}phi(t,x)=f(phi(t,x))$. We know by the chain rule that
                begin{align*}
                frac{partial}{partial{t}}xi(phi(t,x))&=frac{partialxi}{partial{x}}(phi(t,x))cdot frac{partial}{partial{t}}phi(t,x)
                \
                &=frac{partialxi}{partial{x}}(phi(t,x))cdot f(phi(t,x))
                \
                &=frac{partialxi}{partial{x}}(x(t))cdot f(x(t))
                end{align*}



                where $dfrac{partialxi(x)}{partial{x}}cdot{f(x)}$ is the directional derivative of the function $xi$ in the direction of f. This is defined as



                $$dfrac{partialxi(x)}{partial{x}}cdot{f(x)}=nabla{xi(x)}boldsymbol{cdot}f(x)=sum_{n=1}^{N}frac{partial{xi}}{partial{x_i}}f_i(x)$$



                Therefore,



                begin{align*}
                frac{partial}{partial{t}}xi(phi(t,x))&=frac{partialxi}{partial{x}}(x(t))cdot f(x(t))\
                &=sum_{i=1}^n frac{partialxi}{partial{x_i}}(x(t)) f_i(x(t))
                \&=sum_{i=1}^n f_i(phi(t,x))frac{partialxi}{partial{x_i}}(phi(t,x))
                =L_{phi(t,x)}[xi]
                end{align*}



                So along a solution we get $frac{partial}{partial{t}}=L_{phi(t,x)}$. Next, observe that we can write $x(a)$ in series notation as



                $$x(a)=sum_{n=0}^inftyfrac{x^{(n)}}{n!}a^n$$



                We also know that



                $$exp(a)=sum_{n=0}^inftyfrac{a^n}{n!}$$



                Applying $exp(a)$ to the differential operator $afrac{partial}{partial{t}}$ produces



                $$expBig(afrac{partial}{partial{t}}Big)=sum_{n=0}^inftyfrac{a^n}{n!}Big(frac{partial}{partial{t}}Big)^n$$



                Hence if we evaluate this at $x(t)$ we have



                $$expBig(afrac{partial}{partial{t}}Big)x(t)=sum_{n=0}^inftyfrac{a^n}{n!}Big(frac{partial{x(t)}}{partial{t}}Big)^n =sum_{n=0}^inftyfrac{a^n}{n!}x^{n}(t)=sum_{n=0}^inftyfrac{x^{n}(t)}{n!}a^n$$



                Where it is necessary for $t=0$ in order to compute the Maclaurin expansion. Therefore,



                $$expBig(afrac{partial}{partial{t}}Big)x(t)Big|_{t=0}=sum_{n=0}^inftyfrac{x^{n}(t)}{n!}a^n=x(a)$$



                Noting that $x(t)=phi(t,x)$ and changing the constants $arightarrow{t}$, $trightarrow{s}$ so that $x(a)=x(t)$ and $expBig(afrac{partial}{partial{t}}Big)x(t) = expBig(tfrac{partial}{partial{s}}Big)x(s)$ allows us to write



                $$
                phi(t,x)=expleft(tfrac{partial}{partial{s}}right)phi(s,x)Big|_{s=0}
                =expleft(tL_{phi(s,x)}right)[phi(s,x)]Big|_{s=0}
                =expleft(tL_{x}right)[x]
                $$

                If we then replace the exponential with the exponential series, we have
                $$
                phi(t,x)=expleft(tL_{x}right)[x]=Big(sum_{n = 0}^{infty} frac{left(tL_{x}right)^n}{n!}Big)[x]=sum_{n=0}^{infty}frac{t^n}{n!}L^n[x]$$






                share|cite|improve this answer











                $endgroup$



                Here is my interpretation of the first answer.



                Suppose we have a differentiable function $xi:Bbb R^ntoBbb R^n$ where $frac{partial}{partial{t}}phi(t,x)=f(phi(t,x))$. We know by the chain rule that
                begin{align*}
                frac{partial}{partial{t}}xi(phi(t,x))&=frac{partialxi}{partial{x}}(phi(t,x))cdot frac{partial}{partial{t}}phi(t,x)
                \
                &=frac{partialxi}{partial{x}}(phi(t,x))cdot f(phi(t,x))
                \
                &=frac{partialxi}{partial{x}}(x(t))cdot f(x(t))
                end{align*}



                where $dfrac{partialxi(x)}{partial{x}}cdot{f(x)}$ is the directional derivative of the function $xi$ in the direction of f. This is defined as



                $$dfrac{partialxi(x)}{partial{x}}cdot{f(x)}=nabla{xi(x)}boldsymbol{cdot}f(x)=sum_{n=1}^{N}frac{partial{xi}}{partial{x_i}}f_i(x)$$



                Therefore,



                begin{align*}
                frac{partial}{partial{t}}xi(phi(t,x))&=frac{partialxi}{partial{x}}(x(t))cdot f(x(t))\
                &=sum_{i=1}^n frac{partialxi}{partial{x_i}}(x(t)) f_i(x(t))
                \&=sum_{i=1}^n f_i(phi(t,x))frac{partialxi}{partial{x_i}}(phi(t,x))
                =L_{phi(t,x)}[xi]
                end{align*}



                So along a solution we get $frac{partial}{partial{t}}=L_{phi(t,x)}$. Next, observe that we can write $x(a)$ in series notation as



                $$x(a)=sum_{n=0}^inftyfrac{x^{(n)}}{n!}a^n$$



                We also know that



                $$exp(a)=sum_{n=0}^inftyfrac{a^n}{n!}$$



                Applying $exp(a)$ to the differential operator $afrac{partial}{partial{t}}$ produces



                $$expBig(afrac{partial}{partial{t}}Big)=sum_{n=0}^inftyfrac{a^n}{n!}Big(frac{partial}{partial{t}}Big)^n$$



                Hence if we evaluate this at $x(t)$ we have



                $$expBig(afrac{partial}{partial{t}}Big)x(t)=sum_{n=0}^inftyfrac{a^n}{n!}Big(frac{partial{x(t)}}{partial{t}}Big)^n =sum_{n=0}^inftyfrac{a^n}{n!}x^{n}(t)=sum_{n=0}^inftyfrac{x^{n}(t)}{n!}a^n$$



                Where it is necessary for $t=0$ in order to compute the Maclaurin expansion. Therefore,



                $$expBig(afrac{partial}{partial{t}}Big)x(t)Big|_{t=0}=sum_{n=0}^inftyfrac{x^{n}(t)}{n!}a^n=x(a)$$



                Noting that $x(t)=phi(t,x)$ and changing the constants $arightarrow{t}$, $trightarrow{s}$ so that $x(a)=x(t)$ and $expBig(afrac{partial}{partial{t}}Big)x(t) = expBig(tfrac{partial}{partial{s}}Big)x(s)$ allows us to write



                $$
                phi(t,x)=expleft(tfrac{partial}{partial{s}}right)phi(s,x)Big|_{s=0}
                =expleft(tL_{phi(s,x)}right)[phi(s,x)]Big|_{s=0}
                =expleft(tL_{x}right)[x]
                $$

                If we then replace the exponential with the exponential series, we have
                $$
                phi(t,x)=expleft(tL_{x}right)[x]=Big(sum_{n = 0}^{infty} frac{left(tL_{x}right)^n}{n!}Big)[x]=sum_{n=0}^{infty}frac{t^n}{n!}L^n[x]$$







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Jan 7 at 17:01

























                answered Jan 5 at 21:51









                Axion004Axion004

                405413




                405413






























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