Euler Equations

In physics , the Euler Equations describe the rotation of a rigid body in a coordinate system associated with the body itself.

Conclusion

In the reference system of an outside observer, the equations of rotational motion have the form

{\frac {d\mathbf {L} }{dt}}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {d}{dt}}\left(\mathbf {I} \cdot {\boldsymbol {\omega }}\right)=\mathbf {M}

{\ displaystyle {\ frac {d \ mathbf {L}} {dt}} \ {\ stackrel {\ mathrm {def}} {=}} \ {\ frac {d} {dt}} \ left (\ mathbf { I} \ cdot {\ boldsymbol {\ omega}} \ right) = \ mathbf {M}}

In this form, the equations are of little practical use, since, in the general case, both components of the angular momentum — the moment of inertia tensor and the angular velocity pseudovector — depend on time. Euler’s idea was to go into a frame of reference tightly connected with a rotating body. In this system, the moment of inertia tensor is constant, and it can be taken out as a derivative. For further simplification, we choose its main axes of inertia as fixed axes of the body. Thus, we can divide the change in angular momentum into a component that describes the change in magnitude $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ and the component that compensates for this change in direction $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ .

Then the equations take the form:

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

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

Where $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ - the angular momentum of the body with respect to the spatial axes, $\left({\frac {d\mathbf {L} }{dt}}\right)_{\mathrm {relative} }$ ${\ displaystyle \ left ({\ frac {d \ mathbf {L}} {dt}} \ right) _ {\ mathrm {relative}}}$ - a change in the angular momentum of the body with respect to its fixed axes, $\mathbf {\omega }$ ${\ displaystyle \ mathbf {\ omega}}$ the rate of change of the Euler angles of the axes associated with the body with respect to the spatial axes, and $\mathbf {N}$ ${\ displaystyle \ mathbf {N}}$ - external torque.

if we replace $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ its components $I_{1}\omega _{1}\mathbf {e} _{1}+I_{2}\omega _{2}\mathbf {e} _{2}+I_{3}\omega _{3}\mathbf {e} _{3}$ ${\ displaystyle I_ {1} \ omega _ {1} \ mathbf {e} _ {1} + I_ {2} \ omega _ {2} \ mathbf {e} _ {2} + I_ {3} \ omega _ {3} \ mathbf {e} _ {3}}$ then we can replace ${\frac {d\mathbf {L} }{dt}}$ ${\ displaystyle {\ frac {d \ mathbf {L}} {dt}}}$ expression $I_{1}{\dot {\omega }}_{1}\mathbf {e} _{1}+I_{2}{\dot {\omega }}_{2}\mathbf {e} _{2}+I_{3}{\dot {\omega }}_{3}\mathbf {e} _{3}+{\frac {d\mathbf {e} _{1}}{dt}}\omega _{1}I_{1}+{\frac {d\mathbf {e} _{2}}{dt}}\omega _{2}I_{2}+{\frac {d\mathbf {e} _{3}}{dt}}\omega _{3}I_{3}$ ${\ displaystyle I_ {1} {\ dot {\ omega}} _ {1} \ mathbf {e} _ {1} + I_ {2} {\ dot {\ omega}} _ {2} \ mathbf {e} _ {2} + I_ {3} {\ dot {\ omega}} _ {3} \ mathbf {e} _ {3} + {\ frac {d \ mathbf {e} _ {1}} {dt}} \ omega _ {1} I_ {1} + {\ frac {d \ mathbf {e} _ {2}} {dt}} \ omega _ {2} I_ {2} + {\ frac {d \ mathbf {e } _ {3}} {dt}} \ omega _ {3} I_ {3}}$ . if we choose the base vectors $(\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3})$ ${\ displaystyle (\ mathbf {e} _ {1}, \ mathbf {e} _ {2}, \ mathbf {e} _ {3})}$ coinciding with the main axes of inertia of the body, the first three terms are equal $\left({\frac {d\mathbf {L} }{dt}}\right)_{\mathrm {relative} }$ ${\ displaystyle \ left ({\ frac {d \ mathbf {L}} {dt}} \ right) _ {\ mathrm {relative}}}$ and the other three are $\mathbf {\omega } \times \mathbf {L}$ ${\ displaystyle \ mathbf {\ omega} \ times \ mathbf {L}}$ .

Then the Euler equations in component form take the form:

{\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}}

{\ displaystyle {\ 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}}}

It is also possible to use these three equations if the axes in which $\left({\frac {d\mathbf {L} }{dt}}\right)_{\mathrm {relative} }$ ${\ displaystyle \ left ({\ frac {d \ mathbf {L}} {dt}} \ right) _ {\ mathrm {relative}}}$ not related to the body. Then $\mathbf {\omega }$ ${\ displaystyle \ mathbf {\ omega}}$ should be replaced by axis rotation instead of body rotation. However, it is still required that the selected axes be the main axes of inertia! This form of Euler equations is convenient to use for objects with rotational symmetry , which allows you to arbitrarily select some of the main axes of inertia.

Type of equations in an arbitrary local coordinate system

It is possible to choose a local system that does not coincide with the main axes of inertia of the body. In this case, the equations take the form

N_{s}=I_{sq}{\dot {\omega }}_{q}+\varepsilon _{stp}\omega _{t}I_{pq}\omega _{q},

{\ displaystyle N_ {s} = I_ {sq} {\ dot {\ omega}} _ {q} + \ varepsilon _ {stp} \ omega _ {t} I_ {pq} \ omega _ {q},}

Where $I_{sq}$ ${\ displaystyle I_ {sq}}$ is the inertia tensor of the body in the selected local coordinate system.

Euler Equations

Conclusion

Type of equations in an arbitrary local coordinate system

See also

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