Non-canonical transformation, Shankar 2.8.5

[naele]

Homework Statement

Suppose I release a particle at (x=a,y=0) with (p_x = b, p_y = 0) and you release one in the transformed state (x=0, y=a) with (p_x = b, p_y = 0) where the transformation is that we rotate the coordinates but not the momenta. This is a non canonical transformation that leaves H invariant. Show that at later times the states of the two particles are not related by the same transformation.

Homework Equations

The Attempt at a Solution

This problem is a little awkward for me because of ambiguity. Consider a rotation of the coordinates given by
$$x'=x\cos\theta-y\sin\theta,\qquad y'=x\sin\theta+y\cos\theta$$
My initial plan was to compare the solutions to the equations of motion in each coordinate system. In the unprimed system there's no motion in the y-axis due to the initial conditions setting y=0,p_y=0, that is $x(t)=a\cos\omega t+\frac{b}{m\omega}\sin\omega t$. In the primed system using the initial conditions stated $x'(t)=\frac{b}{m\omega}\sin\omega t, y'(t)=a\cos\omega t$

The question asks to show that at later times the two particles are not related by the same transformation, is it referring to the rotation applied to the coordinates and the transformation relating the two solutions?

 

[NateD]

If so, then I can say the solutions are not related by the same transformation since they have different equations, x(t) and x'(t) are not equal.