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Relating to force- General equations of motion

The linéar momentum of a system of particles can also be defined as the product of the total mass  of the system times the velocity of the center of mass

This is commonly known as Newton's second law.

For a more general derivation using tensors, we consider a moving body (see Figure), assumed as a continuum, occupying a volume  at a time , having a surface aréa , with defined traction or surface forces  acting on every point of the body surface, body forces  per unit of volume on every point within the volume , and a velocity field  prescribed throughout the body. Following the previous equation, The linéar momentum of the system is:

By definition the stress vector is  , then

Using the Gauss's divergency theorem to convert a surface integral to a volume integral gives

For an arbitrary volume the integrand itself must be zero, and we have the Cauchy's equation of motion

If a system is in equilibrium, the change in momentum with respect to time is equal to 0, as there is no acceleration.