Euler's equations

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This page discusses rigid body dynamics. For other uses, see Euler function (disambiguation).

In physics, Euler's equations govern the rotation of a rigid body. We choose the body fixed axes to be principal axes of inertia. This will make the calculations easier, since we can now split the change in angular momentum into a component that describes the change of the size of <math>\mathbf{L}</math> and another component that compensates for the change in direction of <math>\mathbf{L}</math>.

The equations are:

<math>

\left(\frac{d\mathbf{L}}{dt}\right)_\mathrm{relative}+\mathbf{\omega}\times\mathbf{L}=\frac{d\mathbf{L}}{dt}=\mathbf{N} </math>

where <math>\mathbf{L}</math> is the projection of the angular momentum in the body fixed axes, <math> \left(\frac{d\mathbf{L}}{dt}\right)_\mathrm{relative}</math> the change of the angular momentum of the body with respect to the body fixed axes, <math>\mathbf{\omega}</math> the vector of angular velocity in the body fixed reference frame, and <math>\mathbf{N}</math> the external torque.

Proof

If we replace <math>\mathbf{L}</math> with its components<math>I_1\omega_1\mathbf{e}_1 + I_2\omega_2\mathbf{e}_2 + I_3\omega_3\mathbf{e}_3</math> we can replace <math>\frac{d\mathbf{L}}{dt}</math> with <math>I_1\dot{\omega}_1\mathbf{e}_1 + I_2\dot{\omega}_2\mathbf{e}_2+I_3\dot{\omega}_3\mathbf{e}_3 + \frac{d\mathbf{e}_1}{dt}\omega_1I_1 + \frac{d\mathbf{e}_2}{dt}\omega_2I_2 + \frac{d\mathbf{e}_3}{dt}\omega_3I_3</math>. If we choose the basis vectors <math>(\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3)</math> to be the body fixed axes, the first three terms are equal to <math>\left(\frac{d\mathbf{L}}{dt}\right)_\mathrm{relative}</math>and the rest is <math>\mathbf{\omega}\times\mathbf{L}</math>

Application

In component form, the Euler equations become

<math>

\begin{matrix} N_1 &=& I_1\dot{\omega}_1+(I_3-I_2)\omega_2\omega_3\\ N_2 &=& I_2\dot{\omega}_2+(I_1-I_3)\omega_3\omega_1\\ N_3 &=& I_3\dot{\omega}_3+(I_2-I_1)\omega_1\omega_2\\ \end{matrix} </math>

For the LHSs equal to zero there are non-trivial solutions: torque-free precession.

It is also possible to use these equations if the axes in which <math> \left(\frac{d\mathbf{L}}{dt}\right)_\mathrm{relative}</math> is described are not connected to the body. <math>\mathbf{\omega}</math> should then be replaced with the rotation of the axes instead of the rotation of the body. It is, however, still required that the chosen axes are still principal axes of inertia! This form of the Euler equations is handy for rotation-symmetric objects that allow some of the principal axes of rotation to be chosen freely.

See Poinsot's construction.ja:オイラーの運動方程式 ru:Уравнения Эйлера (механика)