Equations of Motion

Consider a dynamical system consisting of $N$ particles. Let particle $i$ have mass $m_i$, displacement ${\bf r}_i$, and velocity ${\bf v}_i=d{\bf r}_i/dt$. Suppose that particle $i$ is subject to a force ${\bf f}_{ij}$ exerted by particle $j$. Suppose, in addition, that particle $i$ is subject to an external force (i.e., a force that originates outside the dynamical system) ${\bf F}_i$. Applying Newton's second law of motion to the particle [see Equation (1.19)], we obtain

$$m_i\,\frac{d{\bf v}_i}{dt} = \sum_{j=1,N}^{j\neq i} {\bf f}_{ij} + {\bf F}_i,$$ (1.63)

assuming that all of the forces acting on particle $i$ are superposable. (This is reasonable because gravitational and electromagnetic forces are superposable.) Newton's third law of motion, (1.21), can be generalized to give

$${\bf f}_{ij} = - {\bf f}_{ji},$$ (1.64)

for all $i$ and $j$. Note, in particular, that ${\bf f}_{ii}= -{\bf f}_{ii} = {\bf0}$. In other words, particle $i$ cannot exert a force on itself. This accounts for the exclusion of particle $i$ in the sum on the right-hand side of Equation (1.63).

There are $N$ equations of motion of analogous form to Equation (1.63); one for each particle that makes up the system. We can sum all of these equations to give

$$\sum_{i=1,N}m_i\,\frac{d{\bf v}_i}{dt} =\sum_{i=1,N}\sum_{j=1,N}^{j\neq i} {\bf f}_{ij} + \sum_{i=1,N} {\bf F}_i.$$ (1.65)

Now, every term, ${\bf f}_{ij}$, appearing in the double sum on the right-hand side of the previous equation, can be paired with another term— ${\bf f}_{ji}$, in this case—that is equal and opposite according to Newton's third law of motion, (1.64). In other words, the terms in the sum all cancel out in pairs. It follows that the previous equation reduces to

$$\sum_{i=1,N}m_i\,\frac{d{\bf v}_i}{dt} = {\bf F},$$ (1.66)

where

$${\bf F} = \sum_{i=1,N} {\bf F}_i.$$ (1.67)

is the net external force acting on the system.