from (centripetal) acceleration a(x,y) to x(t) and y(t)
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 and position ) and acceleration . What makes things complicated is that is a function not only of but also of the instantaneous velocity . What's worse, each component of depends on both components of . Any resemblance to centripetal acceleration is not accidental, and if there are methods to "derive" / from for Uniform Circular Motion, they might came in handy here.
Already in a ODE-friendly form, the two components of can be expressed thus ( is a given constant, and and are the components of ):
So the problem lends itself to a direct formulation as an ODE system:
I am no Mathematica wizard, but using a direct application of DSolve I was able to get the following replacement rule:
whose mere application to the equation for yielded an equation that DSolve wasn't able to treat.
Thanks in advance for any pointers on all this. Cheers,