While solving Euler's equation of motion for an asymmetric top, I cam across the three following equations :
$$\dot{\omega_1}+\omega_2 \omega_3=0$$
$$\dot{\omega_2}-\omega_1 \omega_3=0$$
$$\dot{\omega_3}+\frac{\omega_2 \omega_1}{\mu^2}=0$$
The only information that I have been provided is that at time $t=0$, $\omega=N\mu \hat{i}+N\hat{k}$. So I know $\omega_1, \omega_2 $ and $\omega_3$ at time $0$.
How can I decouple and solve the above set of equations ?
I've already been given the solution as follows :
$$\omega_1 =\frac{N\mu}{\cosh(Nt)}$$
$$\omega_2 ={N\mu}\tanh(Nt)$$
$$\omega_3 =\frac{N}{\cosh(Nt)}$$
How do I solve the equations myself ? I've solved linear coupled differential equations using matrices, however this one seems to be nonlinear because of the $\omega_i\omega_j$ terms.
Best Answer
$$\dot{\omega_1}+\omega_2 \omega_3=0$$ $${\omega_1}\dot{\omega_1}+{\omega_1}\omega_2 \omega_3=0 \tag{1} $$ $$\dot{\omega_2}-\omega_1 \omega_3=0$$ $${\omega_2}\dot{\omega_2}-{\omega_2}\omega_1 \omega_3=0 \tag {2}$$ Add both DE and integrate. $${\omega_1}^2+{\omega_2}^2=C_1$$ Third DE gives:
$$\dot{\omega_3}+\frac{\omega_2 \omega_1}{\mu^2}=0$$ $${\omega_3}\dot{\omega_3}+\frac{{\omega_3}\omega_2 \omega_1}{\mu^2}=0$$ $${\omega_3}\dot{\omega_3}-\frac{{\omega_1}\dot{\omega_1}}{\mu^2}=0$$ $${\omega_3}^2-\frac{{\omega_1}^2}{\mu^2}=C_2$$