# How can I approximate Jacobian Elliptic Functions in terms of basic integrable functions for the SO(3) rotation of a rigid body?

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 showed 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 rotatio). 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 than integrate the angular velocity functions to give the position of the body. Does anyone have an idea for a similar method which, just as this one, is not necessarily accurate after a long period of time, but gives a pretty accurate approximation which is enough to approximate the free rotation of a rigid body for a relatively short period of time?

• This may help. An ad hoc method I used many years ago for a similar problem was to simply use these DEs initialized by $\mathrm{sn}(0)=\mathrm{dn}(0)=0$ and $\mathrm{cn}(0)=1$ and integrated them numerically: this works pretty well for small value of $k$. Mathematica has the first kind elliptic integral built in as .... – WetSavannaAnimal Sep 17 '15 at 11:54
• ... $x=\mathrm{EllipticF}[\mathrm{am}(x,\,k) ,\,k^2]$ – WetSavannaAnimal Sep 17 '15 at 11:58
• But the MMOI components are not constant on an inertial frame. – ja72 Apr 2 '16 at 0:27
• No, but the above is for a rotating system. – Bruno KM Apr 4 '16 at 11:18