Hi, all! I am trying to predict the position of an object O under a certain control policy, which defines what acceleration should be applied to O for every point in space. All I have is the start conditions (velocity $\displaystyle v_0$ and position $\displaystyle p_0$) and acceleration $\displaystyle a(x,y)$. What makes things complicated is that $\displaystyle a(x,y)$ is a function not only of $\displaystyle (x,y)$ but also of the instantaneous velocity $\displaystyle v(t)$. What's worse, each component of $\displaystyle a(t)$ depends on both components of $\displaystyle v(t)$. Any resemblance to centripetal acceleration is not accidental, and if there are methods to "derive" $\displaystyle x(t)$/$\displaystyle y(t)$ from $\displaystyle a(t)$ for Uniform Circular Motion, they might came in handy here.

Already in a ODE-friendly form, the two components of $\displaystyle a(t)$ can be expressed thus ($\displaystyle r$ is a given constant, and $\displaystyle x'(t)$ and $\displaystyle y'(t)$ are the components of $\displaystyle v(t)$):

$\displaystyle a_x(t) = \frac{2 y'(t) \left(\sqrt{r^2-y(t)^2} y'(t)-y(t) x'(t)\right)}{r^2}$

$\displaystyle a_y(t) = \frac{2 x'(t) \left(y(t) x'(t)-\sqrt{r^2-y(t)^2} y'(t)\right)}{r^2}$

So the problem lends itself to a direct formulation as an ODE system:

$\displaystyle x''(t) - a_x(t) = 0$

$\displaystyle y''(t) - a_y(t) = 0$

I am no Mathematica wizard, but using a direct application of DSolve I was able to get the following replacement rule:

$\displaystyle x'(t) \rightarrow e^{-\frac{y(t)^2}{r^2}} \int_1^t \frac{2 e^{\frac{y(K[1])^2}{r^2}} \sqrt{r^2-y(K[1])^2} y'(K[1])^2}{r^2} \, dK[1]+c_1 e^{-\frac{y(t)^2}{r^2}}$,

whose mere application to the equation for $\displaystyle y(t)$ yielded an equation that DSolve wasn't able to treat.

Thanks in advance for any pointers on all this. Cheers,

Jorge.