Non-isolated systems

(2.38) |

where is the internal force exerted by object on object , and the external force acting on object .

The equation of motion of the th object is

Summing over all objects, we obtain

(2.40) |

which reduces to

where

(2.42) |

is the net external force acting on the system. Here, the sum over the internal forces has cancelled out in pairs as a result of Newton's third law of motion. (See Section 2.5.) We conclude that if there is a net external force acting on the system then the total linear momentum evolves in time according to Equation (2.41) but is completely unaffected by any internal forces. The fact that Equation (2.41) is similar in form to Equation (2.13) suggests that the center of mass of a system consisting of many point objects has analogous dynamics to a single point object, whose mass is the total system mass, moving under the action of the net external force.

Taking Equation (2.39), and summing over all objects, we obtain

where

(2.44) |

is the net external torque (about the origin) acting on the system. Here, the sum over the internal torques has cancelled out in pairs, assuming that the internal forces are central in nature. (See Section 2.5.) We conclude that if there is a net external torque acting on the system then the total angular momentum evolves in time according to the simple equation (2.43) but is completely unaffected by any internal torques.