# 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:

$\left(\begin{matrix} {\bold F} \\ {\boldsymbol \tau} \end{matrix}\right) = \left(\begin{matrix} m {\boldsymbol 1} & 0 \\ 0 & {\bold I}_{\rm cm} \end{matrix}\right) \left(\begin{matrix} \bold a_{\rm cm} \\ {\boldsymbol \alpha} \end{matrix}\right) + \left(\begin{matrix} 0 \\ {\boldsymbol \omega} \times {\bold I}_{\rm cm} \, {\boldsymbol \omega} \end{matrix}\right),$

where

F = total force acting on the center of mass
m = mass of the body
1 = 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:

$\left(\begin{matrix} {\bold F} \\ {\boldsymbol \tau}_{\rm p} \end{matrix}\right) = \left(\begin{matrix} m {\boldsymbol 1} & - m [{\bold c}]^{\times}\\ m [{\bold c}]^{\times} & {\bold I}_{\rm cm} - m[{\bold c}]^{\times}[{\bold c}]^{\times}\end{matrix}\right) \left(\begin{matrix} \bold a_{\rm p} \\ {\boldsymbol \alpha} \end{matrix}\right) + \left(\begin{matrix} m[{\boldsymbol \omega}]^{\times}[{\boldsymbol \omega}]^{\times} {\bold c} \\ {[\boldsymbol \omega]}^\times ({\bold I}_{\rm cm} - m [{\bold c}]^\times[{\bold c}]^\times)\, {\boldsymbol \omega} \end{matrix}\right),$

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

$[\mathbf{c}]^{\times} \equiv \left(\begin{matrix} 0 & -c_z & c_y \\ c_z & 0 & -c_x \\ -c_y & c_x & 0 \end{matrix}\right) \qquad \qquad [\mathbf{\boldsymbol{\omega}}]^{\times} \equiv \left(\begin{matrix} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{matrix}\right)$

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

$\left(\begin{matrix} m {\boldsymbol 1} & - m [{\bold c}]^{\times}\\ m [{\bold c}]^{\times} & {\bold I}_{\rm cm} - m [{\bold c}]^{\times}[{\bold c}]^{\times}\end{matrix}\right),$

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

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

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]