Example how to ﬁnd Lie group (x,y) given canonical coordinates R,S

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

\begin{align*} x & =R\\ y & =SR \end{align*}

Hence

\begin{align*} \bar {x} & =\bar {R}\\ \bar {y} & =\bar {S}\bar {R}\end{align*}

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

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

Using the values given for \(R,S\) 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*}