Example how to ﬁnd Lie group (x,y) given canonical coordinates X,Y

Given\(\ X=x,Y=\frac {y}{x}\) ﬁnd Lie group \(\bar {x},\bar {y}\). Solving for \(x,y\) from \(X,Y\) gives

\begin{align*} x & =X\\ y & =YX \end{align*}

Hence

\begin{align*} \bar {x} & =\bar {X}\\ \bar {y} & =\bar {Y}\bar {X}\end{align*}

But \(\bar {Y}=Y+\epsilon \) by deﬁnition of canonical coordinates and \(\bar {X}=X\) by deﬁnition of canonical coordinates. Hence the above becomes

\begin{align*} \bar {x} & =X\\ \bar {y} & =\left ( Y+\epsilon \right ) X \end{align*}

Using the values given for \(X,Y\) in terms of \(x,y\) the above becomes

\begin{align*} \bar {x} & =x\\ \bar {y} & =\left ( \frac {y}{x}+\epsilon \right ) x\\ & =y+\epsilon x \end{align*}