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:
- 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 that is not coincident with the center of mass, the equations assume the more complex form:
where c is the location of the center of mass, and
The inertial terms are contained in the spatial inertia matrix
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.
See also 
- 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.
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