I am trying to derive a Lie bracket, and then find the related integral curve at the point $(x_0,y_0)$.
The problem gives the vector fields $X = y \frac{\partial }{\partial x}$ ,$Y = \frac{x^2}{2} \frac{\partial }{\partial y}$. The goal is then to find $[X,Y] = XY-YX$, which is the Lie bracket (and another vector field) and then find the integral curve through $(x_0,y_0)$.
I've tried applying the definition of the Lie bracket as follows:
$[X,Y]= y\frac{\partial\frac{x^2}{2}}{\partial x}\frac{\partial }{\partial x}-\frac{x^2}{2}\frac{\partial y}{\partial y}\frac{\partial }{\partial y}$, from which you can simplify and get: $yx\frac{\partial }{\partial x} -\frac{x^2}{2}\frac{\partial }{\partial y}$.
I then proceed to get the integral curve, but don't quite get what is listed as a solution. I try listing this as a system of two ODE's (i.e $x'(t) = yx, y'(t) =\frac{x^2}{2}$, but once I finish solving, I don't get the result ($x(t)= \frac{2x_0}{x_0t+2}$, $y(t) = \frac{1}{4}y_0(x_0t+2)^2$.
Can anyone see where I've may have made any mistakes?
Best Answer
The error is when you compute the Lie bracket: $$[X,Y]=y\frac\partial{\partial x}\left(\frac{x^2}2\frac\partial{\partial y}\right)-\frac{x^2}2\frac\partial{\partial y}\left(y\frac\partial{\partial x}\right)=yx\frac\partial{\partial y}-\frac{x^2}2\frac\partial{\partial x}.$$
Notice that you have the vector fields wrong (you inverted $x$ and $y$).
Then you have to solve: $$\begin{cases}x'=-x^2/2\\y'=xy.\end{cases}$$ Integrating $x'=-x^2/2$ first yields $$x(t)=\frac{2x_0}{x_0t+2}$$ and substituting in $y'=xy$ and solving yields $$y=\frac{y_0}4(x_0t+2)^2.$$