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I am having serious problems in deriving the coordinate component expression for Lie derivative of vector fields.

I already know how to do that using an outdated coordinate-based approach mostly used in old physics literature, and I also know I can do this in a more simple way by using a coordinate system adapted to my vector field, I want to get the correct form using the definition of the Lie derivative.

Some of this might be because of my low understanding of flows.

Let $M$ be a real, $n$-dimensional, $C^\infty$ manifold, let $X$ and $V$ be smooth vector fields defined in the neighborhood of a point $p\in M$, and let $ (U,x) $ be a chart so that $p\in U$ and $x(p)=(x^1(p),...,x^n(p))$.

Let be $V=V^\mu \partial/\partial x^\mu$ and $X=X^\mu\partial/\partial x^\mu$ in $U$.

Let $\Phi^X_t$ be the flow of $X$. If I understand this well, then $$ \frac{d}{dt}\Phi^X_t(p)=X(p). $$

The Lie-derivative should be $$ \left.\mathcal{L}_XV\right|_p=\frac{d}{dt}(\Phi^X_{-t})_*(\left.V\right|_{\Phi^X_t(p)}). $$

My attempt was as follows: $$ \mathcal{L}_XV|_p[f]=\frac{d}{dt}(V|_{\Phi^X_t(p)}[f\circ\Phi^X_{-t}])= \\ =\frac{d}{dt}(V|_{\Phi^X_t(p)}[f\circ x^{-1}\circ x\circ\Phi^X_{-t}]), $$ then, I try to evaluate the vector field on that massive composition by $$ \frac{d}{dt}(V^\mu(\Phi^X_t)\left.\frac{\partial}{\partial x^\mu}\right|_{x(\Phi^X_t)}(f\circ x^{-1}\circ x\circ\Phi^X_{-t}))= \\ =\frac{d}{dt}(V^\mu(\Phi^X_t)\left.\frac{\partial}{\partial x^\mu}\right|_{x(\Phi^X_{-t})}(f\circ x^{-1})\frac{\partial}{\partial x^\nu}(x\circ\Phi^X_{-t})), $$ and then what? I am not even sure that where I wrote $\partial/\partial x^\nu$, should I have written $d/dt$ instead? In either case, I have no idea how to go any further, and any help is greatly appreciated.

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1 Answer 1

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In the meantime, I managed to solve this, I am posting my solution here in case someone is struggling with the same thing.

Let $M$ be a real, $C^\infty$, $n$-dimensional manifold, let $X$ and $V$ be $C^\infty$ vector field defined in the neighborhood of some point $p$ and let $(U,x)$ be a chart with $p\in U$. The coordinate expression for $X$ and $V$ are $$ X_p=X^\mu(p)\left.\frac{\partial}{\partial x^\mu}\right|_p\ \mathrm{and}\ V_p=V^\mu(p)\left.\frac{\partial}{\partial x^\mu}\right|_p. $$ All vector fields from now on are evaluated at point $p$ with $p$ being reasonable arbitrary, so I will omit the $p$'s where it is beneficial.

One thing to note that the coordinate representation of the flow $\Phi^X_t(p)$, $x\circ\Phi^X_t\circ x^{-1}$ at $t=0$, and only at $t=0$ can be expressed as $$ x^\mu\circ\Phi^X_t\circ x^{-1}=x^\mu+tX^\mu, $$ since at $t=0$, this gives back the identity map, and its derivative at $t=0$ are the components of $X$.

Then, $$ \mathcal{L}_XV=\lim_{t\rightarrow0}\frac{(\Phi^X_{-t})_*(V\circ\Phi^X_t)-V}{t}= \\ \ \\ =\lim_{t\rightarrow0}\frac{(\Phi^X_{-t*})^\mu_{\ \ \nu}(V^\nu\circ\Phi^X_t)-V^\mu}{t}\frac{\partial}{\partial x^\mu}, $$ now I switch to using the above expression for the flow, since we'll be taking the limit at $t=0$, and also note that the matrix of the pushforward via $\Phi^X_{-t}$ at $(x+tX)$ (since we are moving from coordinates $x+tX$ to $x$) is $$ \left.\frac{\partial(x^\mu-tX^\mu)}{\partial x^\nu}\right|_{x+tX}=\delta^\mu_\nu-t\left.\frac{\partial X^\mu}{\partial x^\nu}\right|_{x+tX}, $$ and so $$ \mathcal{L}_XV=\lim_{t\rightarrow0}\frac{(\delta^\mu_\nu-t\left.\frac{\partial X^\mu}{\partial x^\nu}\right|_{x+tX})V^\nu(x+tX)-V^\mu(x)}{t}\frac{\partial}{\partial x^\mu}= \\ \ \\ =\lim_{t\rightarrow0}\frac{V^\mu(x+tX)-V^\mu(x)-t\left.\frac{\partial X^\mu}{\partial x^\nu}\right|_{x+tX}V^\nu(x+tX)}{t}\frac{\partial}{\partial x^\mu}= \\ \ \\=\lim_{t\rightarrow0}\left[\frac{V^\mu(x+tX)-V^\mu(x)}{t}-\left.\frac{\partial X^\mu}{\partial x^\nu}\right|_{x+tX}V^\nu(x+tX)\right]\frac{\partial}{\partial x^\mu}= \\ \ \\ \left(X^\nu\frac{\partial V^\mu}{\partial x^\nu}-V^\nu\frac{\partial X^\mu}{\partial x^\nu}\right)\frac{\partial}{\partial x^\mu}, $$ where everything in the last line is evaluated at $p$ or the coordinates of $p$.

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