Euler's equations (rigid body dynamics)

Euler's equations (rigid body dynamics)

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

In physics, Euler's equations describe the rotation of a rigid body in a frame of reference fixed in the rotating body

:egin{matrix}I_1dot{omega}_{1}+(I_3-I_2)omega_2omega_3 &=& M_{1}\I_2dot{omega}_{2}+(I_1-I_3)omega_3omega_1 &=& M_{2}\I_3dot{omega}_{3}+(I_2-I_1)omega_1omega_2 &=& M_{3}end{matrix}

where M_{k} are the applied torques, I_{k} are the principal moments of inertia and omega_{k} are the components of the angular velocity vector oldsymbolomega along the principal axes.

Motivation and derivation

In absolute space, i.e., in a "space-fixed" frame of reference, the time derivative of angular momentum equals the applied torque

:frac{dmathbf{L{dt} stackrel{mathrm{def{=} frac{d}{dt} left( mathbf{I} cdot oldsymbolomega ight) = mathbf{M}

where mathbf{I} is the moment of inertia tensor. Although this law is universally true, it is not always helpful in solving for the motion of a general rotating rigid body, since both mathbf{I} and oldsymbolomega can change during the motion.

Therefore, we change to a coordinate frame fixed in the rotating body, and chosen so that its axes are aligned with the principal axes of the moment of inertia tensor. In this frame, at least the moment of inertia tensor is constant (and diagonal), which simplifies calculations. As described in the moment of inertia, the angular momentum vector mathbf{L} can be written

:mathbf{L} stackrel{mathrm{def{=} L_{1}mathbf{e}_{1} + L_{2}mathbf{e}_{2} + L_{3}mathbf{e}_{3} = I_{1}omega_{1}mathbf{e}_{1} + I_{2}omega_{2}mathbf{e}_{2} + I_{3}omega_{3}mathbf{e}_{3}

where the I_{k} are the principal moments of inertia, the mathbf{e}_{k} are unit vectors in the directions of the principal axes, and the omega_{k} are the components of the angular velocity vector along the principal axes. In a "rotating" reference frame, the time derivative must be replaced with

:left(frac{dmathbf{L{dt} ight)_mathrm{rot}+oldsymbolomega imesmathbf{L}=mathbf{M}

where the rot subscript indicates that it is taken in the rotating reference frame. Substituting L_{k} stackrel{mathrm{def{=} I_{k}omega_{k}, taking the cross product and using the fact that the principal moments do not change with time, we arrive at the Euler equations

:egin{matrix}I_1dot{omega}_{1}+(I_3-I_2)omega_2omega_3&=&M_{1}\I_2dot{omega}_{2}+(I_1-I_3)omega_3omega_1&=&M_{2}\I_3dot{omega}_{3}+(I_2-I_1)omega_1omega_2&=&M_{3}end{matrix}

Torque-free solutions

For the RHSs equal to zero there are non-trivial solutions: torque-free precession. Notice that if I is constant (because the inertia tensor is the identity, because we work in the intrinsecal frame, or because the torque is driving the rotation around the same axis mathbf{hat{n so that mathrm{I} is not changing) then we may write

:mathbf{M} stackrel{mathrm{def{=} I frac{domega}{dt}mathbf{hat{n = I alpha mathbf{hat{n

where:alpha is called the "angular acceleration" (or "rotational acceleration") about the rotation axis mathbf{hat{n.

But if I is not constant in the external reference frame (ie. the body is moving and its inertia tensor is not the identity) then we cannot take the I outside the derivate. In this cases we will have torque-free precession, in such a way that I(t) and w(t) change together so that their derivative is zero. This motion can be visualized by Poinsot's construction.

Generalizations

It is also possible to use these equations if the axes in which left(frac{dmathbf{L{dt} ight)_mathrm{relative} is described are not connected to the body. oldsymbolomega 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.

ee also

* Moment of inertia
* Poinsot's construction

References

* Landau LD and Lifshitz EM (1976) "Mechanics", 3rd. ed., Pergamon Press. ISBN 0-08-021022-8 (hardcover) and ISBN 0-08-029141-4 (softcover).
* Goldstein H. (1980) "Classical Mechanics", 2nd. ed., Addison-Wesley. ISBN 0-201-02918-9
* Symon KR. (1971) "Mechanics", 3rd. ed., Addison-Wesley. ISBN 0-201-07392-7


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