# Euler's Equations of Motion for Rotation of Rigid Body

Jump to navigation
Jump to search

## 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

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## 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**