Rotational Invariance

The displacement, ${\bf r}_i$, is, by definition, a vector (i.e., its components, $x_i$, $y_i$, $z_i$, transform under rotation of the coordinate axes in such a manner that the length and direction of ${\bf r}_i$ are preserved). (See Section A.5.) Moreover, $m_i$ and $t$ are scalars (i.e., they are invariant under rotation of the coordinate axes). It follows that ${\bf v}_i=d{\bf r}_i/dt$ and $d{\bf v}_i/dt$ are vectors. Furthermore, we have already seen that forces are vectors. (See Section 1.2.3.) Finally, we know that if ${\bf a}$ and ${\bf b}$ are vectors then so is ${\bf a}\times {\bf b}$. (See Section A.8.) It follows that every term appearing in the previous two equations transforms as a vector under rotation of the coordinate axes. In other words, the forms of the linear and angular equations of motion, (1.89) and (1.90), respectively, are invariant under rotation of the coordinate axes. Of course, this must be the case because the choice of the orientation of the axes of a Cartesian coordinate system is completely arbitrary, and has no bearing on the motions of bodies in the universe.