Euler's Equations of Motion for Rotation of Rigid Body
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Theorem
Let a rigid body $B$ rotate about an axis $\AA$ which is fixed in relation to $B$ and parallel to the principal axis of inertia of $B$.
Then the rotation of $B$ about $\AA$ is described by:
- $\mathbf I \cdot \dot {\boldsymbol \omega} + \boldsymbol \omega \times \paren {\mathbf I \cdot \boldsymbol\omega} = \mathbf M$
where:
- $\mathbf M$ is the torque applied to $B$ about $\AA$
- $\mathbf I$ is the moment of inertia of $B$ with respect to $\AA$
- $\boldsymbol \omega$ is the angular velocity about $\AA$.
Proof
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Also presented as
Euler's Equations of Motion for Rotation of Rigid Body can also be seen presented in the form:
\(\ds I_1 \dfrac {\partial \omega_1} {\partial t} - \paren {I_2 - I_3} \omega_2 \omega_3\) | \(=\) | \(\ds M_1\) | ||||||||||||
\(\ds I_2 \dfrac {\partial \omega_2} {\partial t} - \paren {I_3 - I_1} \omega_3 \omega_1\) | \(=\) | \(\ds M_2\) | ||||||||||||
\(\ds I_3 \dfrac {\partial \omega_3} {\partial t} - \paren {I_1 - I_2} \omega_1 \omega_2\) | \(=\) | \(\ds M_3\) |
where:
- $I_1$, $I_2$ and $I_1$ are the components of the torque applied about the principal axes
- $I_1$, $I_2$ and $I_1$ are the moments of inertia at fixed point $O$
- $\omega_1$, $\omega_2$ and $\omega_3$ are the components of angular velocity along the principal axis.
Source of Name
This entry was named for Leonhard Paul Euler.
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3$: Appendix $\text A$: Euler
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.21$: Euler ($\text {1707}$ – $\text {1783}$)
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Euler's equations
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Euler's equations