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${\displaystyle {\textbf {F}}={\frac {d}{dt}}(m{\textbf {v}})}$ Second law of motion
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In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with angular velocity ω whose axes are fixed to the body. Their general vector form is

${\displaystyle \mathbf {I} {\dot {\boldsymbol {\omega }}}+{\boldsymbol {\omega }}\times \left(\mathbf {I} {\boldsymbol {\omega }}\right)=\mathbf {M} .}$

where M is the applied torques and I is the inertia matrix. The vector ${\displaystyle {\dot {\boldsymbol {\omega }}}}$ is the angular acceleration. Again, note that all quantities are defined in the rotating reference frame.

In orthogonal principal axes of inertia coordinates the equations become

${\displaystyle {\begin{aligned}I_{1}\,{\dot {\omega }}_{1}+(I_{3}-I_{2})\,\omega _{2}\,\omega _{3}&=M_{1}\\I_{2}\,{\dot {\omega }}_{2}+(I_{1}-I_{3})\,\omega _{3}\,\omega _{1}&=M_{2}\\I_{3}\,{\dot {\omega }}_{3}+(I_{2}-I_{1})\,\omega _{1}\,\omega _{2}&=M_{3}\end{aligned}}}$

where M_k are the components of the applied torques, I_k are the principal moments of inertia and ω_k are the components of the angular velocity.

In the absence of applied torques, one obtains the Euler top. When the torques are due to gravity, there are special cases when the motion of the top is integrable.

Derivation

In an inertial frame of reference (subscripted "in"), Euler's second law states that the time derivative of the angular momentum L equals the applied torque:

${\displaystyle {\frac {d\mathbf {L} _{\text{in}}}{dt}}=\mathbf {M} _{\text{in}}}$

For point particles such that the internal forces are central forces, this may be derived using Newton's second law. For a rigid body, one has the relation between angular momentum and the moment of inertia I_in given as

${\displaystyle \mathbf {L} _{\text{in}}=\mathbf {I} _{\text{in}}{\boldsymbol {\omega }}}$

In the inertial frame, the differential equation is not always helpful in solving for the motion of a general rotating rigid body, as both I_in and ω can change during the motion. One may instead change to a coordinate frame fixed in the rotating body, in which the moment of inertia tensor is constant. Using a reference frame such as that at the center of mass, the frame's position drops out of the equations. In any rotating reference frame, the time derivative must be replaced so that the equation becomes

${\displaystyle \left({\frac {d\mathbf {L} }{dt}}\right)_{\mathrm {rot} }+{\boldsymbol {\omega }}\times \mathbf {L} =\mathbf {M} }$

and so the cross product arises, see time derivative in rotating reference frame. The vector components of the torque in the inertial and the rotating frames are related by ${\displaystyle \mathbf {M} _{\text{in}}=\mathbf {Q} \mathbf {M} ,}$ where ${\displaystyle \mathbf {Q} }$ is the rotation tensor (not rotation matrix), an orthogonal tensor related to the angular velocity vector by ${\displaystyle {\boldsymbol {\omega }}\times {\boldsymbol {u}}={\dot {\mathbf {Q} }}\mathbf {Q} ^{-1}{\boldsymbol {u}}}$ for any vector u. Now ${\displaystyle \mathbf {L} =\mathbf {I} {\boldsymbol {\omega }}}$ is substituted and the time derivatives are taken in the rotating frame, while realizing that the particle positions and the inertia tensor does not depend on time. This leads to the general vector form of Euler's equations which are valid in such a frame

${\displaystyle \mathbf {I} {\dot {\boldsymbol {\omega }}}+{\boldsymbol {\omega }}\times \left(\mathbf {I} {\boldsymbol {\omega }}\right)=\mathbf {M} .}$

The equations are also derived from Newton's laws in the discussion of the resultant torque.

More generally, by the tensor transform rules, any rank-2 tensor ${\displaystyle \mathbf {T} }$ has a time-derivative ${\displaystyle \mathbf {\dot {T}} }$ such that for any vector ${\displaystyle \mathbf {u} }$, one has ${\displaystyle \mathbf {\dot {T}} \mathbf {u} ={\boldsymbol {\omega }}\times (\mathbf {T} \mathbf {u} )-\mathbf {T} ({\boldsymbol {\omega }}\times \mathbf {u} )}$. This yields the Euler's equations by plugging in ${\displaystyle {\frac {d}{dt}}\left(\mathbf {I} {\boldsymbol {\omega }}\right)=\mathbf {M} .}$

Special case solutions

Generalized Euler equations

The Euler equations can be generalized to any simple Lie algebra.^[1] The original Euler equations come from fixing the Lie algebra to be ${\displaystyle {\mathfrak {so}}(3)}$, with generators ${\displaystyle {t_{1},t_{2},t_{3}}}$ satisfying the relation ${\displaystyle [t_{a},t_{b}]=\epsilon _{abc}t_{k}}$. Then if ${\displaystyle {\boldsymbol {\omega }}(t)=\sum _{a}\omega _{a}(t)t_{a}}$ (where ${\displaystyle t}$ is a time coordinate, not to be confused with basis vectors ${\displaystyle t_{a}}$) is an ${\displaystyle {\mathfrak {so}}(3)}$-valued function of time, and ${\displaystyle \mathbf {I} =\mathrm {diag} (I_{1},I_{2},I_{3})}$ (with respect to the Lie algebra basis), then the (untorqued) original Euler equations can be written

$${\displaystyle \mathbf {I} {\dot {\boldsymbol {\omega }}}=[\mathbf {I} {\boldsymbol {\omega }},{\boldsymbol {\omega }}].}$$

To define ${\displaystyle \mathbf {I} }$ in a basis-independent way, it must be a self-adjoint map on the Lie algebra ${\displaystyle {\mathfrak {g}}}$ with respect to the invariant bilinear form on ${\displaystyle {\mathfrak {g}}}$. This expression generalizes readily to an arbitrary simple Lie algebra, say in the standard classification of simple Lie algebras.

This can also be viewed as a Lax pair formulation of the generalized Euler equations, suggesting their integrability.

See also

• Euler angles
• Dzhanibekov effect
• Moment of inertia
• Poinsot's ellipsoid
• Rigid rotor

References

• C. A. Truesdell, III (1991) A First Course in Rational Continuum Mechanics. Vol. 1: General Concepts, 2nd ed., Academic Press. ISBN 0-12-701300-8. Sects. I.8-10.
• C. A. Truesdell, III and R. A. Toupin (1960) The Classical Field Theories, in S. Flügge (ed.) Encyclopedia of Physics. Vol. III/1: Principles of Classical Mechanics and Field Theory, Springer-Verlag. Sects. 166–168, 196–197, and 294.
• Landau L.D. and Lifshitz E.M. (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