Newton–Euler equations
In classical mechanics, the Newton–Euler equations describe the combined translational and rotational dynamics of a rigid body.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.With respect to a coordinate frame whose origin coincides with the body's center of mass for τ(torque) and an inertial frame of reference for F(force), they can be expressed in matrix form as: where 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 vector from P to the center of mass of the body expressed in the body-fixed frame, and denote skew-symmetric cross product matrices.The inertial terms are contained in the spatial inertia matrix while the fictitious forces are contained in the term:[6] 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.