Classical mechanics, Hamiltonian formalism, change of variables

[c299792458]

Homework Statement

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 [itex]\theta[/itex] be some parameter.

And
[tex]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
[/tex]

Suppose the original Hamiltonian is [tex]H(x,y)={1\over 2}(x_1^2+y_1^2+x_2^2+y_2^2)[/tex]
I wish to find solve for the motion in terms of the new variables. I am also given the restriction that [itex]X_2=Y_2=0[/itex]


____
***Attempt:***

I believe we have [tex]H(X,Y)={1\over 2}(X_1^2+Y_1^2+X_2^2+Y_2^2)[/tex]



Now the normal Hamiltonian formalism would suggest that [tex]\dot X_i={\partial H\over \partial Y_i }\\
\dot Y_i=-{\partial H\over \partial X_i }[/tex]

Which gives [tex]\ddot X_1=-X_1\\
\ddot Y_1=-Y_1[/tex]
Therefore, [tex]X_1(t)=A(\theta)\cos t+B(\theta)\sin t\\
Y_2(t)=C(\theta)\cos t+D(\theta)\sin t[/tex]***Is this form of solutions right?***

We see that the [tex]{\partial X_1\over \partial \theta}=-Y_2=0\\
{\partial Y_1\over \partial \theta}=X_2=0[/tex]
So [itex]A,B,C,D[/itex] must be constants.

Are these arguments right? And can I get a better solution, say by getting a more specific set of [itex]A,B,C,D[/itex], given only the given information?

Thank you.

 

[lippyka]

Homework Equations \dot X_i={\partial H\over \partial Y_i }\\\dot Y_i=-{\partial H\over \partial X_i }The Attempt at a SolutionYes, your arguments and solution are correct. In order to get more specific values for A,B,C,D, you would need more information about the system, such as initial conditions or other constraints.