Newton–Euler equations


In classical mechanics, the Newton–Euler equations describe the combined translational and rotational dynamics of a rigid body.
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 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:
where

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:
where c is the location of the center of mass expressed in the body-fixed frame,
and
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
while the fictitious forces are contained in the term:
When the center of mass is not coincident with the coordinate frame, the translational and angular accelerations 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 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.