Conservation of Linear Momentum

Suppose that our dynamical system is isolated. In other words, the system is not subject to a net external force, so that ${\bf F}={\bf0}$. In this case, Equation (1.66) reduces to

$$\sum_{i=1,N}m_i\,\frac{d{\bf v}_i}{dt} = {\bf0}.$$ (1.73)

However, the linear momentum of the $i$th particle is

$${\bf p}_i = m_i\,{\bf v}_i.$$ (1.74)

Thus, Equation (1.73) yields

$$\sum_{i=1,N}\frac{d{\bf p}_i}{dt} = {\bf0},$$ (1.75)

or

$$\frac{d{\bf P}}{dt} = {\bf0},$$ (1.76)

where

$${\bf P} = \sum_{i=1,N} {\bf p}_i$$ (1.77)

is the total linear momentum of the system. Equation (1.76) implies that the total linear momentum of an isolated dynamical system is a conserved quantity. In other words, the total momentum does not evolve in time.