How can I find phase angle based on mouse movement?

Question

It's me again with my ball problematic, after 3 closed questions that really helped me out I figured out I would make a short video, so you'll have a better understanding of the problem. I have this formula:

\begin{eqnarray} x(t) &=& A e^{-\gamma t/2} \cos(\omega t - \phi) \\ y(t) &=& B e^{-\gamma t/2} \cos(\omega t - \psi)\\ \end{eqnarray}

this formula is for an unforced 2d damped harmonic oscillator and honestly I've tested it and is looks like it's the good one. Now my problem is for $\phi$ and $\psi$. When I set them to $0$ I get an oscillation on a straight line but as you can see in the video I suspect that based on the mouse movement appears a phase shift on either one or two of the axis making the ball rotate around the equilibrium. At this point I'm unable to understand this motion and reproduce it neither knowing which forces are implied.

the description of the motion is the following:

• When I click on the screen the ball gets attracted tho the clicked point
• When I hold the mouse pressed the ball oscillates around the clicked point (equilibrium)
• When I release the mouse, the ball continues it's way based on the direction and velocity of the oscillation
• When reaching the end of the object the ball collides and I guess I only need to change the sign of $dt$ (where $dt$ is the difference between current frame position and previous one)
• When moving the mouse I guess operates some sort of centripetal force which I'm unable to emulate at this point.

Here is the link to the video

my question is how can I find both phase angles to emulate this motion.

Answer 1

Your damped equations in x and y are basically related to forces: $$ \frac{d^2\vec{r}}{d^2t}=\vec{F}/m, $$ where $$\vec{r}=(x,y)$$ in your case. Also the force is 2D. The two dimensions are independent, and it is quite easy to "integrate" your x,y formulas to see the actual forces. By clicking the mouse you create a force and can calculate all the parameters of your formulas (as you did alread, I think).

So what happens when you move the mouse while pressing the button?

You have to add up forces. It cannot be a single force, but only the sum of at least two (not parallel) forces that produce the phase shift. Just consider, while you move the mouse, you create a stream of forces, each with a different direction, magnitude -- and each one is damped in time, so the latest one is always dominating:

$$ \frac{d^2\vec{r}}{d^2t}=\frac{1}{m} \sum_i \vec{F_i} $$

If you like, just consider two independent perpendicular forces/movements, add them up, and observe the result.