# Newton–Euler equations

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:

${\displaystyle \left({\begin{\mathbf }\\{\boldsymbol {\tau }}\end}\right)=\left({\beginm{\mathbf _}&0\\0&{\mathbf }_{\rm }\end}\right)\left({\begin\mathbf _{\rm }\\{\boldsymbol {\alpha }}\end}\right)+\left({\begin0\\{\boldsymbol {\omega }}\times {\mathbf }_{\rm }\,{\boldsymbol {\omega }}\end}\right),}$

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:

${\displaystyle \left({\begin{\mathbf }\\{\boldsymbol {\tau }}_{\rm }\end}\right)=\left({\beginm{\mathbf _}&-m[{\mathbf }]^{\times }\\m[{\mathbf }]^{\times }&{\mathbf }_{\rm }-m[{\mathbf }]^{\times }[{\mathbf }]^{\times }\end}\right)\left({\begin\mathbf _{\rm }\\{\boldsymbol {\alpha }}\end}\right)+\left({\beginm[{\boldsymbol {\omega }}]^{\times }[{\boldsymbol {\omega }}]^{\times }{\mathbf }\\{[{\boldsymbol {\omega }}]}^{\times }({\mathbf }_{\rm }-m[{\mathbf }]^{\times }[{\mathbf }]^{\times })\,{\boldsymbol {\omega }}\end}\right),}$

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

${\displaystyle [\mathbf ]^{\times }\equiv \left({\begin0&-c_&c_\\c_&0&-c_\\-c_&c_&0\end}\right)\qquad \qquad [\mathbf {\boldsymbol {\omega }} ]^{\times }\equiv \left({\begin0&-\omega _&\omega _\\\omega _&0&-\omega _\\-\omega _&\omega _&0\end}\right)}$

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

${\displaystyle \left({\beginm{\mathbf _}&m[{\mathbf }]^{\times }\\m[{\mathbf }]^{\times }&{\mathbf }_{\rm }-m[{\mathbf }]^{\times }[{\mathbf }]^{\times }\end}\right),}$

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

${\displaystyle \left({\beginm{[{\boldsymbol {\omega }}]}^{\times }{[{\boldsymbol {\omega }}]}^{\times }{\mathbf }\\{[{\boldsymbol {\omega }}]}^{\times }({\mathbf }_{\rm }-m[{\mathbf }]^{\times }[{\mathbf }]^{\times })\,{\boldsymbol {\omega }}\end}\right).}$

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]