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Currently Playing: All power-law force has a conformal dual

Given any power-law force, we can conformally transform to some other power-law force (except for inverse cube law). What does that mean? Next video: [Not done] Previous video: : https://www.youtube.com/watch?v=1ubTR3HzAI4&list=PLDcSwjT2BF_XHbQkuyPnfDmb5jSlALLB3&index=2 This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject. Files for download: Go to https://www.mathemaniac.co.uk/download and enter the following password: conformallydualforces Sources: Bohlin’s wiki (only in Swedish): https://sv.wikipedia.org/wiki/Karl_Bohlin Bohlin’s portrait: https://commons.wikimedia.org/wiki/File:Karl_Bohlin_SPA10_(cropped).jpg Original Bohlin paper: https://www.persee.fr/doc/bastr_0572-7405_1911_num_28_1_12893 Arnol’d and Vasil’ev paper (at the end section): https://www.jstor.org/stable/24099031?seq=4 Arnold’s book “Huygens and Barrow, Newton and Hooke” (see appendix 1): https://physicsgg.me/wp-content/uploads/2020/07/huygens-and-barrow-newton-and-hooke-vladimir-i.-arnold.pdf Kasner’s colloquium publication (pp. 81- 84): https://ia800603.us.archive.org/2/items/differentialgeom00kasn/differentialgeom00kasn.pdf Arnold’s portrait: https://commons.wikimedia.org/wiki/File:Vladimir_Arnold-1.jpg Vasil’ev’s portrait: https://commons.wikimedia.org/wiki/File:Victor_Vassiliev.jpg Further reading: Tristan Needham’s account: https://www.jstor.org/stable/2323768 (I find this even more complicated than the argument I presented, so I didn't use the argument in this paper, and haven't really read it in too much detail either. I realised the day before publishing this video that this part was also in Visual Complex Analysis.) Video chapters: 00:00 Introduction 00:51 The big picture 02:16 Recap of last video 04:20 Applying the formula 08:25 "Isosceles triangle" argument 11:34 Combining lengths for numerator 12:55 Computing denominator 16:04 Generalisation 21:09 Two final remarks Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels: https://forms.gle/QJ29hocF9uQAyZyH6 If you want to know more interesting Mathematics, stay tuned for the next video! SUBSCRIBE and see you in the next video! If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos. Social media: Facebook: https://www.facebook.com/mathemaniacyt Instagram: https://www.instagram.com/_mathemaniac_/ Twitter: https://twitter.com/mathemaniacyt Patreon: https://www.patreon.com/mathemaniac (support if you want to and can afford to!) Merch: https://mathemaniac.myspreadshop.co.uk Ko-fi: https://ko-fi.com/mathemaniac [for one-time support] For my contact email, check my About page on a PC. See you next time!