Every Power-Law Force Has a Conformal Dual

Explore the fascinating mathematical relationship between power-law forces and conformal transformations in this 23-minute educational video. Discover how any power-law force can be conformally transformed into another power-law force, with the notable exception of the inverse cube law. Learn about the theoretical foundations through a recap of conformal mapping concepts, then dive into practical applications of the transformation formula. Follow along with geometric arguments including the "isosceles triangle" approach, detailed calculations for combining lengths in the numerator, and comprehensive computation of the denominator. Examine the generalization of these principles and understand two important final remarks that complete the mathematical framework. The presentation builds upon previous work in the series while showcasing this underrated mathematical topic with novel approaches, supported by historical sources including Bohlin's original paper and Arnold's mathematical contributions.