About
I found a video like the following on YouTube: it captured the behavior of a rigid body rotating in outer space. Watching it, you can see the object rotating around a certain axis, but at some point suddenly, it appears to abruptly change its axis of rotation. This happens despite being in outer space, an environment where no external forces are acting on it.
What kind of theoretical aspect is behind ?
I happened to find an explanation of it on a book, “Particla Method, theory of SPH, MPS and DEM”. In order not to forget the things, I will summarize it briefly here.
The rotation of a rigid bodey
Using the equations of motion, we consider the cross product with position. This can then be described using the total angular momentum L.
\sum_{i=1}^{N} (x_i \times \frac{d}{dt}p_i)\\
=\frac{d}{dt}\sum_{i=1}^{N} (x_i \times p_i) = \frac{d}{dt}\sum_{i=1}^{N} (l_i)=\frac{d}{dt}Land this form is equals to summation of dot product position x_i with force vector f_i. It means that torque.
\begin{align}
\frac{d}{dt}L=\tau \\
\end{align}\\
\text{where } \tau=\sum_{i=1}^{N} (x_i \times f_i)It says that this (1) is equivalent to motion equation. In essence, it is basically just a difference between expressing the change in momentum over time in terms of rotational components or in terms of linear motion.
Therefore, we can express angular momentum 𝐿 using the moment of inertia 𝐼 and angular velocity just as momentum P is expressed as the product of mass M and velocity V, P=MV.
L=I\omega
The value of inertia tensor is defined by atitude of rigid body. It means that it is a function which is decided with respect to time t.
I_{ii}=\int_{V}\rho(x_j^2+x_k^2)dV\\
I_{jk}=I_{kj}=-\int_{V}\rho x_2x_3dVThe moment of inertia is regarded as resistance with respect to rotation. If it is small, the body can rotate rapidly with a certain moment.
Why does the object rotate in a complex manner
In an environment where no external forces act, such as in space, the angular momentum of a rigid body is conserved.
L=const.
However, the moment of inertia is a quantity that changes depending on the orientation of the object, and in some cases, it is not as simple as a diagonal matrix.
I asked about it to ChatGPT and got good answer.
In my simple summary, it can be described as follows: The moment of inertia has three principal moments of inertia, I_1, I_2, and I_3 , and the stability differs depending on how the rotation axis aligns with these moments. The cause of instability arises when the rotation axis aligns with the intermediate axis (corresponding to I_2 ), where small perturbations can have a significant impact on the angular velocity vector \boldsymbol{\omega}, leading to oscillations or reversals.
