Question

A particle in a central field moves in the spiral orbit 𝑟 = 𝑐𝜃3

Answer 1

Using the orbit equation

$$L=mr^2\frac{d\theta}{dt}=mc^2\theta^6\frac{d\theta}{dt}$$

This can be integrated to give 

$$Lt = \frac{1}{7}mc^2\theta ^7$$