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${\displaystyle {\textbf {F}}={\frac {d}{dt}}(m{\textbf {v}})}$ Second law of motion
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In classical mechanics, the Newton–Euler equations describe the combined translational and rotational dynamics of a rigid body.^{[1][2] [3][4][5]}

Traditionally the Newton–Euler equations is the grouping together of Euler's two laws of motion for a rigid body into a single equation with 6 components, using column vectors and matrices. These laws relate the motion of the center of gravity of a rigid body with the sum of forces and torques (or synonymously moments) acting on the rigid body.

Center of mass frame

With respect to a coordinate frame whose origin coincides with the body's center of mass for τ(torque) and an inertial frame of reference for F(force), they can be expressed in matrix form as:

${\displaystyle \left({\begin{matrix}{\mathbf {F} }\\{\boldsymbol {\tau }}\end{matrix}}\right)=\left({\begin{matrix}m{\mathbf {I} _{3}}&0\\0&{\mathbf {I} }_{\rm {cm}}\end{matrix}}\right)\left({\begin{matrix}\mathbf {a} _{\rm {cm}}\\{\boldsymbol {\alpha }}\end{matrix}}\right)+\left({\begin{matrix}0\\{\boldsymbol {\omega }}\times {\mathbf {I} }_{\rm {cm}}\,{\boldsymbol {\omega }}\end{matrix}}\right),}$

where

F = total force acting on the center of mass

m = mass of the body

I₃ = the 3×3 identity matrix

a_cm = acceleration of the center of mass

v_cm = velocity of the center of mass

τ = total torque acting about the center of mass

I_cm = moment of inertia about the center of mass

ω = angular velocity of the body

α = angular acceleration of the body

Any reference frame

With respect to a coordinate frame located at point P that is fixed in the body and not coincident with the center of mass, the equations assume the more complex form:

${\displaystyle \left({\begin{matrix}{\mathbf {F} }\\{\boldsymbol {\tau }}_{\rm {p}}\end{matrix}}\right)=\left({\begin{matrix}m{\mathbf {I} _{3}}&-m^{\times }\\m^{\times }&{\mathbf {I} }_{\rm {cm}}-m^{\times }{\mathbf {c} }^{\times }\end{matrix}}\right)\left({\begin{matrix}\mathbf {a} _{\rm {p}}\\{\boldsymbol {\alpha }}\end{matrix}}\right)+\left({\begin{matrix}m{\boldsymbol {\omega }}^{\times }{\boldsymbol {\omega }}^{\times }{\mathbf {c} }\\{{\boldsymbol {\omega }}}^{\times }({\mathbf {I} }_{\rm {cm}}-m{\mathbf {c} }^{\times }{\mathbf {c} }^{\times })\,{\boldsymbol {\omega }}\end{matrix}}\right),}$

where c is the vector from P to the center of mass of the body expressed in the body-fixed frame, and

${\displaystyle \mathbf {c} ^{\times }\equiv \left({\begin{matrix}0&-c_{z}&c_{y}\\c_{z}&0&-c_{x}\\-c_{y}&c_{x}&0\end{matrix}}\right)\qquad \qquad \mathbf {\boldsymbol {\omega }} ^{\times }\equiv \left({\begin{matrix}0&-\omega _{z}&\omega _{y}\\\omega _{z}&0&-\omega _{x}\\-\omega _{y}&\omega _{x}&0\end{matrix}}\right)}$

denote skew-symmetric cross product matrices.

The left hand side of the equation—which includes the sum of external forces, and the sum of external moments about P—describes a spatial wrench, see screw theory.

The inertial terms are contained in the spatial inertia matrix

Zdroj:https://en.wikipedia.org?pojem=Newton–Euler_equations
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