So, the rotation of a 3d body can be described with Euler's equations of motion giving the rotational velocity in components along the principal axes of inertia. As shown in f.ex. this paper,
Euler Top (free asymmetric top): solution of Euler’s equations in terms of elliptic integrals, Berry Groisman, Cambridge University, 2014.
they can be expressed in terms of Jacobian Elliptic Functions sn, cn and dn (in case of torque-free rotation). I tried to approximate these functions using the trigonometric functions, such that: $$ \operatorname{sn}(x,k)=\sin(\operatorname{am}(x,k))≈\sin\left(\frac{\sin(2u)}{2} C(k)+u\right) $$ where $u=x\times(\pi/\text{Period of }\operatorname{sn}(x,k))$ and $C(k)$ is a function of $k$ which matches how "wide" the sin graph is at the top and is constant for a given body so that u is the only variable.
However, I cannot find a way to translate these component functions of angular velocity from along the principal axes (body frame) to the inertial frame of reference and then integrate the angular velocity functions to give the position of the body. Does anyone have an idea for a similar method which, just like this one, is not necessarily accurate after a long period, but gives a pretty accurate approximation which is enough to approximate the free rotation of a rigid body for a relatively short period?
