where \(\left\{\bullet,\bullet\right\}\) are the Poisson brackets with respect to the variables \(\left(q,p\right)\). This condition is indeed quite general. However, the only allowed transformation from (\ref{25c}) to the equations (\ref{eq:02a}) is given by \(x=q\), and \(y=e^{-\dot{G}t/2}p\) provided \(G\) is homogeneous of degree zero. This is the case of constant \(g\) that will be addressed below. In this case, (\ref{25c}) are equivalent to (\ref{eq:02a}) only for constant \(g\).
Constant parameters
Let us analyze the case \(g^2 \le \omega^2\) with both \(\omega\) and \(g\) constant parameters, and \(F=F(t)\) still arbitrary. In this case, the function \(G\) should be linear in \(t\). Let us suppose it to have the form of \(G=2gt\). We also have the solution