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${\displaystyle \mathbf{\text{F}}=\frac{d\mathbf{p}}{dt}}$ Second law of motion
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Inclassical mechanics,Euler's rotation equations are a vectorial quasilinearfirst-order ordinary differential equation describing the rotation of arigid body, using arotating reference frame withangular velocity ω whose axes are fixed to the body. They are named in honour ofLeonhard Euler.

In the absence of appliedtorques, one obtains theEuler top. When the torques are due togravity, there are special cases when the motion of the top isintegrable.

Their general vector form is

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

whereM is the appliedtorques andI is theinertia matrix.The vector${\displaystyle \dot{\mathit{\omega }}}$ is theangular acceleration. Again, note that all quantities are defined in the rotating reference frame.

Inorthogonal principal axes of inertia coordinates the equations become

whereM_k are the components of the applied torques,I_k are theprincipal moments of inertia and ω_k are the components of the angular velocity.

In aninertial frame of reference (subscripted "in"),Euler's second law states that thetime derivative of theangular momentumL equals the appliedtorque:

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

For point particles such that the internal forces arecentral forces, this may be derived usingNewton's second law.For a rigid body, one has the relation between angular momentum and themoment of inertiaI_in given as

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

In the inertial frame, the differential equation is not always helpful in solving for the motion of a general rotating rigid body, as bothI_in andω 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

${\displaystyle {\left(\frac{d\mathbf{L}}{dt}\right)}_{\mathrm{r}\mathrm{o}\mathrm{t}}+\mathit{\omega }\times \mathbf{L}=\mathbf{M}}$

and so the cross product arises, seetime 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 (notrotation matrix), anorthogonal tensor related to the angular velocity vector by${\displaystyle \mathit{\omega }\times \mathit{u}=\dot{\mathbf{Q}}{\mathbf{Q}}^{-1}\mathit{u}}$for any vectoru.Now${\displaystyle \mathbf{L}=\mathbf{I}\mathit{\omega }}$ 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

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

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

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

When choosing a frame so that its axes are aligned with the principal axes of the inertia tensor, its component matrix is diagonal, which further simplifies calculations. As described in themoment of inertia article, the angular momentumL can then be written

${\displaystyle \mathbf{L}=L_1{\mathbf{e}}_1+L_2{\mathbf{e}}_2+L_3{\mathbf{e}}_3=\sum _{i=1}^3{I}_i{\omega }_i{\mathbf{e}}_i}$

Also in some frames not tied to the body can it be possible to obtain such simple (diagonal tensor) equations for the rate of change of the angular momentum. Thenω must be the angular velocity for rotation of that frames axes instead of the rotation of the body. It is however still required that the chosen axes are still principal axes of inertia. The resulting form of the Euler rotation equations is useful for rotation-symmetric objects that allow some of the principal axes of rotation to be chosen freely.

Torque-freeprecessions are non-trivial solution for the situation where the torque on theright hand side is zero. WhenI is not constant in the external reference frame (i.e. the body is moving and its inertia tensor is not constantly diagonal) thenI cannot be pulled through thederivative operator acting onL. In this caseI(t) andω(t) do change together in such a way that the derivative of their product is still zero. This motion can be visualized byPoinsot's construction.

The Euler equations can be generalized to anysimple Lie algebra.^[1] The original Euler equations come from fixing the Lie algebra to be${\displaystyle \mathfrak{s}\mathfrak{o}(3)}$, with generators${\displaystyle t_1,t_2,t_3}$ satisfying the relation${\displaystyle [t_a,t_b]={ϵ}_{abc}{t}_c}$. Then if${\displaystyle \mathit{\omega }(t)=\sum _a{\omega }_a(t)t_a}$ (where${\displaystyle t}$ is a time coordinate, not to be confused with basis vectors${\displaystyle t_a}$) is an${\displaystyle \mathfrak{s}\mathfrak{o}(3)}$-valued function of time, and${\displaystyle \mathbf{I}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(I_1,I_2,I_3)}$ (with respect to the Lie algebra basis), then the (untorqued) original Euler equations can be written^[2]$${\displaystyle \mathbf{I}\dot{\mathit{\omega }}=[\mathbf{I}\mathit{\omega },\mathit{\omega }].}$$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 theinvariant 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 aLax pair formulation of the generalized Euler equations, suggesting their integrability.

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

• 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) andISBN 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

• Rigid bodies
• Rigid bodies mechanics
• Rotation in three dimensions
• Equations

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