# Solving for position of a SO(3) rotating object, given the integrable functions for components of angular velocity along the principle axes

Assuming that you have approximated or solved the Euler's Equations for components of angular velocity along its principal axes of inertia $x$, $y$ and $z$ - i.e. in the coordinate system that is rotating with the body- in the torque free case, and that the resulting functions $\omega_x$, $\omega_y$, $\omega_z$ of time for these components are integrable, how would you than go about solving the function for position of the body in the inertial coordinate system $X$, $Y$ and $Z$? I have tried using Euler's angles and to solve for the precession, nutation and spin; however, I failed at separating the terms (splitting spins, precessions and nutations into three equations). I would sincerely appreciate if someone could present how they would solve these equations, or offer a different solution (using f.ex. quaternions or rotation matrix) that would give the solution for the rotational position of the body as a function of time (or an approximation, that is still a function of time).

Sidenote:

I am not highly interested in computing the answer: I would really like to find the function of time, even if it is only an approximate.

• The equations of motion of rigid, rotating bodies are not integrable. There is no closed form solution in the general case (which us chaotic). If you want to know the solutions for special cases you will have to wade trough thousands of pages of textbooks and papers on the topic. People have been attacking this problem literally for the past two hundred years, or so. As is the question is way too broad. – CuriousOne Sep 22 '15 at 18:52
• I am aware of that; however, I stated the assumption that the function for angular velocity components is given. The function itself is usually the problem that has been tacled for centuries, but again, I'm assuming it is known – Bruno KM Sep 22 '15 at 19:30
• I see. Would it be easier to tackle this with a rotation matrix? Even then there seems to be a need for integration of implicit functions. I don't know if there is a closed form solution for that. – CuriousOne Sep 22 '15 at 20:00