Conservation of Angular Momentum

Taking the vector product of Equation (1.63) with the displacement ${\bf r}_i$, we obtain

$$m_i\,{\bf r}_i\times \frac{d{\bf v}_i}{dt} = \sum_{j=1,N}^{j\neq i} {\bf r}_i\times {\bf f}_{ij} + {\bf r}_i\times {\bf F}_i.$$ (1.78)

Now, the angular momentum of particle $i$ about the origin of the coordinate system is defined

$${\bf l}_i = {\bf r}_i\times {\bf p}_i = m_i\,{\bf r}_i\times {\bf v}_i.$$ (1.79)

(See Section A.8.) It follows that

$$\frac{d{\bf l}_i}{dt} = m_i\,{\bf v}_i\times {\bf v}_i + m_i\,{\bf r}_i\times \frac{d{\bf v}_i}{dt} = m_i\,{\bf r}_i\times \frac{d{\bf v}_i}{dt}.$$ (1.80)

(See Sections A.8 and A.12.) Hence, Equation (1.78) yields the following angular equation of motion for the $i$th particle:

$$\frac{d{\bf l}_i}{dt} = \sum_{j=1,N}^{j\neq i} {\bf r}_i\times {\bf f}_{ij} + {\bf r}_i\times {\bf F}_i.$$ (1.81)

There are $N$ angular equations of motion of analogous form to the previous equation; one for each particle that makes up the system. We can sum all of these equations to give

$$\frac{d{\bf L}}{dt} =\sum_{i=1,N} \sum_{j=1,N}^{j\neq i} {\bf r}_i\times {\bf f}_{ij} + \mbox{\boldmath$\tau$} ,$$ (1.82)

where

$${\bf L} = \sum_{i=1,N} {\bf l}_i$$ (1.83)

is the total angular momentum of the system about the origin of the coordinate system, and

$$\mbox{\boldmath$\tau$} = \sum_{i=1,N} {\bf r}_i\times {\bf F}_i$$ (1.84)

is the net external torque acting on the system about the origin of the coordinate system. (See Section A.8.)

Consider the double sum on the right-hand side of Equation (1.82). A general term, ${\bf r}_i\times {\bf f}_{ij}$, in this sum can always be paired with a matching term, ${\bf r}_j\times {\bf f}_{ji}$, in which the indices have been swapped. Making use of Equation (1.64), the sum of a general matched pair can be written

$${\bf r}_i\times {\bf f}_{ij}+{\bf r}_j\times {\bf f}_{ji} = ({\bf r}_i-{\bf r}_j)\times {\bf f}_{ij}.$$ (1.85)

Let us assume that the forces acting between the various components of the system are central in nature, so that ${\bf f}_{ij}$ is parallel to ${\bf r}_i-{\bf r}_j$. In other words, the force exerted on particle $j$ by particle $i$ either points directly toward, or directly away from, particle $i$, and vice versa. This is a reasonable assumption, because most of the forces that we encounter in the world around us are of this type (e.g., gravity). It follows that if the forces are central in nature then the vector product on the right-hand side of the previous expression is zero. (See Section A.8.) We conclude that

$${\bf r}_i\times {\bf f}_{ij}+{\bf r}_j\times {\bf f}_{ji} = {\bf0},$$ (1.86)

for all values of $i$ and $j$. Thus, the double sum on the right-hand side of Equation (1.82) is zero for any kind of central internal force. We are left with

$$\frac{d {\bf L}}{dt} = \mbox{\boldmath$\tau$} .$$ (1.87)

In particular, if the system is isolated, such that it is not subject to a net external torque, so that $\tau$$={\bf0}$, then the previous equation reduces to

$$\frac{d {\bf L}}{dt} = {\bf0}.$$ (1.88)

In other words, the total angular momentum of an isolated system is a conserved quantity, provided that the different components of the system interact via central forces.