# Newton–Euler equations

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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, 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
acm = acceleration of the center of mass
vcm = velocity of the center of mass
τ = total torque acting about the center of mass
Icm = 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:

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

where c is the location of the center of mass expressed in the body-fixed frame, and

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

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

${\displaystyle \left({\begin{matrix}m{{\mathbf {I}}_{3}}&-m[{\mathbf {c}}]^{\times }\\m[{\mathbf {c}}]^{\times }&{\mathbf {I}}_{\rm {cm}}-m[{\mathbf {c}}]^{\times }[{\mathbf {c}}]^{\times }\end{matrix}}\right),}$

while the fictitious forces are contained in the term:[6]

${\displaystyle \left({\begin{matrix}m{[{\boldsymbol {\omega }}]}^{\times }{[{\boldsymbol {\omega }}]}^{\times }{\mathbf {c}}\\{[{\boldsymbol {\omega }}]}^{\times }({\mathbf {I}}_{\rm {cm}}-m[{\mathbf {c}}]^{\times }[{\mathbf {c}}]^{\times })\,{\boldsymbol {\omega }}\end{matrix}}\right).}$

When the center of mass is not coincident with the coordinate frame (that is, when c is nonzero), the translational and angular accelerations (a and α) are coupled, so that each is associated with force and torque components.

## Applications

The Newton–Euler equations are used as the basis for more complicated "multi-body" formulations (screw theory) that describe the dynamics of systems of rigid bodies connected by joints and other constraints. Multi-body problems can be solved by a variety of numerical algorithms.[2][6][7]