# Euler’s equation for rigid bodies

Let $1$ be an inertial frame body (a rigid body) and $2$ a rigid body in motion respect to an observer located at $1$. Let $Q$ be an arbitrary point (fixed or in motion) and $C$ the center of mass of $2$. Then,

 $\displaystyle\mathbf{M}_{Q}=\boldsymbol{\mathbb{I}}^{Q}\boldsymbol{\alpha}_{21% }+\boldsymbol{\omega}_{21}\times(\boldsymbol{\mathbb{I}}^{Q}\boldsymbol{\omega% }_{21})+m\mathbf{\overline{QC}}\times\mathbf{a}^{Q2}_{1},$ (1)

where $m$ is the mass of the rigid body, $\mathbf{\overline{QC}}$ the position vector of $C$ respect to $Q$, $\mathbf{M}_{Q}$ is the moment of forces system respect to $Q$, $\boldsymbol{\mathbb{I}}^{Q}$ the tensor of inertia respect to orthogonal axes embedded in $2$ and origin at $Q2$ 11That is possible because the kinematical concept of frame extension., and $\mathbf{a}^{Q2}_{1}$, $\boldsymbol{\omega}_{21}$, $\boldsymbol{\alpha}_{21}$, are the acceleration of $Q2$, the angular velocity and acceleration vectors respectively, all of them measured by an observer located at $1$.
This equation was got by Euler by using a fixed system of principal axes with origin at $C2$. In that case we have $Q=C$, and therefore

 $\displaystyle\mathbf{M}_{C}=\boldsymbol{\mathbb{I}}^{C}\boldsymbol{\alpha}_{21% }+\boldsymbol{\omega}_{21}\times(\boldsymbol{\mathbb{I}}^{C}\boldsymbol{\omega% }_{21}).$ (2)

Euler used three independent scalar equations to represent (2). It is well known that the number of degrees of freedom associate to a rigid body in free motion in $\mathbb{R}^{3}$ are six, just equal the number of independent scalar equations necessary to solve such a motion. (Newton’s law contributing with three)
Its is clear if $2$ is at rest or in uniform and rectilinear translation, then $\mathbf{M}_{Q}=\mathbf{0}$, one of the necessary and sufficient conditions for the equilibrium of the system of forces applied to a rigid body. (The other one is the force resultant $\mathbf{F}=\mathbf{0}$)

Title Euler’s equation for rigid bodies EulersEquationForRigidBodies 2013-03-22 17:10:36 2013-03-22 17:10:36 perucho (2192) perucho (2192) 8 perucho (2192) Topic msc 70G45