This problem has to do with a canonical transformation and Hamiltonian formalism. Below is my attempt at solving it, but I am not too sure about it. Please help!
Let $\theta$ be some parameter.
And $X_1=x_1\cos \theta-y_2\sin\theta\\ Y_1=y_1\cos \theta+x_2\sin\theta\\ X_2= x_2\cos \theta-y_1\sin\theta\\ Y_2=y_2\cos\theta+x_1\sin \theta $
Suppose the original Hamiltonian is $H(x,y)={1\over 2}(x_1^2+y_1^2+x_2^2+y_2^2)$ I wish to find solve for the motion in terms of the new variables. I am also given the restriction that $X_2=Y_2=0$
Attempt:
I believe we have $H(X,Y)={1\over 2}(X_1^2+Y_1^2+X_2^2+Y_2^2)$
Now the normal Hamiltonian formalism would suggest that $\dot X_i={\partial H\over \partial Y_i }\\ \dot Y_i=-{\partial H\over \partial X_i }$
Which gives $\ddot X_1=-X_1\\ \ddot Y_1=-Y_1$ Therefore, $X_1(t)=A(\theta)\cos t+B(\theta)\sin t\\ Y_2(t)=C(\theta)\cos t+D(\theta)\sin t$*Is this form of solutions right?*
We see that the ${\partial X_1\over \partial \theta}=-Y_2=0\\ {\partial Y_1\over \partial \theta}=X_2=0$ So $A,B,C,D$ must be constants.
Are these arguments right? And can I get a better solution, say by getting a more specific set of $A,B,C,D$, given only the given information?
Thank you.