I have the following problem:
We have the discrete dynamical system $\Phi: \mathbb{R}^4-\{q_1^2=q_2^2\}\mapsto\mathbb{R}^4$ given by:
$\tilde{q_1}=p_1(q_1^2+q_2^2)+2q_1q_2p_2$
$\tilde{q_2}=p_2(q_1^2+q_2^2)+2q_1q_2p_1$
$\tilde{p_1}=\dfrac{q_1}{q_1^2-q_2^2}$
$\tilde{q_2}=-\dfrac{q_2}{q_1^2-q_2^2}$
Prove that the functions
$F_1(q_1,q_2,p_1,p_2)=q_1p_1+q_2p_2$, $F_2(q_1,q_2,p_1,p_2)=q_1p_2+q_2p_1$
are integrals of motion.
Now consider the change of coordinates $\Psi:(q_1,q_2,p_1,p_2)\mapsto(Q_1,Q_2,P_1,P_2)$
$Q_1=\dfrac{1}{2}(\log(q_1+q_2)+\log(q_1-q_2))$
$Q_2=\dfrac{1}{2}(\log(q_1+q_2)-\log(q_1-q_2))$
$P_1=q_1p_1+q_2p_2$
$P_2=q_1p_2+q_2p_1$.
Prove that $\Psi$ allows to linearize $\Phi$:
$\tilde{Q_1}=Q_1+v_1(F_1,F_2)$
$\tilde{Q_2}=Q_2+v_1(F_1,F_2)$
$\tilde{P_1}=P_1$
$\tilde{P_2}=P_2$.
$v_1,v_2$ need to be determined.
This is basically the question and I allready showed that $F_1,F_2$ are integrals of motion but I have now idea for the second part. I dont now, what I need to show and how this linearization of the system is defined. All help and hints are apprechiated, thanks!