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