dU/dτ= -X -αU + βV
dV/dτ= -Y -αV - βU
The initial conditions are X=1 and Y=0 and the puck is motionless.
Here is something to try. Consider various values of β, say ranging from 0.2 to 4, and take α=0. Note the puck does not enter a "forbidden" circle around the origin. For higher values of β, the forbidden circle is larger. Can derive an analytical expression for the radius of the circle? Also note that for larger values of β that the kinetic energy remains small and most of the potential energy is never converted to potential energy.