Appell's Equation of Motion - Example: Euler's Equations

Example: Euler's Equations

Euler's equations provide an excellent illustration of Appell's formulation.

Consider a rigid body of N particles joined by rigid rods. The rotation of the body may be described by an angular velocity vector, and the corresponding angular acceleration vector


\boldsymbol\alpha = \frac{d\boldsymbol\omega}{dt}

The generalized force for a rotation is the torque N, since the work done for an infinitesimal rotation is . The velocity of the kth particle is given by


\mathbf{v}_{k} = \boldsymbol\omega \times \mathbf{r}_{k}

where rk is the particle's position in Cartesian coordinates; its corresponding acceleration is


\mathbf{a}_{k} = \frac{d\mathbf{v}_{k}}{dt} =
\boldsymbol\alpha \times \mathbf{r}_{k} + \boldsymbol\omega \times \mathbf{v}_{k}

Therefore, the function S may be written as


S = \frac{1}{2} \sum_{k=1}^{N} m_{k} \left( \mathbf{a}_{k} \cdot \mathbf{a}_{k} \right)
= \frac{1}{2} \sum_{k=1}^{N} m_{k} \left\{ \left(\boldsymbol\alpha \times \mathbf{r}_{k} \right)^{2}
+ \left( \boldsymbol\omega \times \mathbf{v}_{k} \right)^{2}
+ 2 \left( \boldsymbol\alpha \times \mathbf{r}_{k} \right) \cdot \left(\boldsymbol\omega \times \mathbf{v}_{k}\right) \right\}

Setting the derivative of S with respect to equal to the torque yields Euler's equations


I_{xx} \alpha_{x} - \left( I_{yy} - I_{zz} \right)\omega_{y} \omega_{z} = N_{x}

I_{yy} \alpha_{y} - \left( I_{zz} - I_{xx} \right)\omega_{z} \omega_{x} = N_{y}

I_{zz} \alpha_{z} - \left( I_{xx} - I_{yy} \right)\omega_{x} \omega_{y} = N_{z}

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