2.2.22.12 Example how to find Lie group \(\left ( \bar {x},\bar {y}\right ) \) 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*}