Newton–Euler equations

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

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 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

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

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.

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]

• Euler's laws of motion for a rigid body.
• Euler angles
• Inverse dynamics
• Centrifugal force
• Principal axes
• Spatial acceleration
• Screw theory of rigid body motion.