Eulers equations
lastmod 2024-03-05created 2020-09-201 edits
Euler's equations of motion are equations that describe the motion of a rigid body. They are named after Leonhard Euler.
Definition
Suppose a torque \(\tau\) is applied to a rigid body with one point fixed in three-dimensional space. For convenience, choose a Cartesian coordinate system (Korean: 직교좌표계) along the eigenbasis of the moment of inertia tensor. That is, the moment of inertia tensor has the following form.
\[ \mathsf I=\begin{pmatrix} I_1\\&I_2\\&&I_3 \end{pmatrix} \]
Then the motion of the rigid body satisfies the following three equations, called Euler's equations of motion.
\[ \tau_1=\dot L_1=I_1\dot\omega_1+(I_3-I_2)\omega_2\omega_3 \]
\[ \tau_2=\dot L_2=I_2\dot\omega_2+(I_1-I_3)\omega_3\omega_1 \]
\[ \tau_3=\dot L_3=I_3\dot\omega_3+(I_2-I_1)\omega_1\omega_2 \] .
where
- \(\mathbf L=(L_1,L_2,L_3)\) is the angular momentum of the rigid body
- \(\omega=(\omega_1,\omega_2,\omega_3)\) is the angular velocity of the rigid body
- \(\tau=(\tau_1,\tau_2,\tau_3)\) is the torque applied to the rigid body.