Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 11

Direct substitution in the Möbius transformation givesNeither substitution meets the excluded denominator point $z=-i$.

For real $\lambda$, the numerator and denominator have equal complex modulus:The denominator is always nonzero. HenceThus the real axis maps into the unit circle in the complex plane.

Differentiate the polar form of a complex number using the product rule and chain rule:A second differentiation givesThe two contributions to $2\dot{r}\dot{\theta }$ arise separately from differentiating $r\dot{\theta }$ and from differentiating the moving exponential factor.

Write $z=x+iy$. Squaring the nonnegative complex modulus on each side givesThe linear terms cancel, leaving $3(x^2+y^2)=12$. Therefore the locus isIts real and imaginary axis intersections are $\pm 2$ and $\pm 2i$.

On this circle the radial coordinate is constant, $r=2$, so $\dot{r}=\ddot{r}=0$. The velocity is a tangent vector to the circle in the anticlockwise direction, represented by $i{e}^{i\theta }$ when $\dot{\theta }>0$. Substituting into part (b),The inward component is the centripetal acceleration; the transverse component allows the speed to change. Anticlockwise motion does not imply uniform speed or $\ddot{\theta }=0$. The preceding figure includes the required tangent arrow.

The perpendicular unit vectors represented by $e^{i\theta }$ and $i{e}^{i\theta }$ point radially outwards and transversely in the increasing-angle direction. Reading off their real coefficients in part (b) gives the signed componentsThese are velocity in polar coordinates and acceleration in polar coordinates. In particular $r\dot{\theta }$ is a transverse speed component; $\dot{\theta }$ alone is an angular velocity. The moving basis, rather than any extra force assumption, accounts for the additional acceleration terms. At the origin the polar directions themselves are undefined, although the Cartesian velocity and acceleration can remain well defined.