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Currently Playing: The hidden conservation law of gravity

The LRL vector seems like a mystery - it's straightforward to show that it is conserved in gravity, but how did one come up with it in the first place? Here is a derivation first documented in 2008 using Bohlin transformation / Arnold-Vasiliev duality and complex numbers, and uses the "complex" analogue of energy in harmonic oscillators to deduce the LRL vector in gravity. 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: lrliscomplexenergy Sources: (1) From action principle to Maupertuis’s principle: (a) V. Arnold “Mathematical Methods of Classical Mechanics” pp. 243-246 (b) Landau and Lifshitz “Mechanics” pp. 140-142 (2) using Maupertuis’s principle for this duality (both papers did it): (a) Arnol’d and Vasil’ev paper (at the end): https://www.jstor.org/stable/24099031?seq=4 (b) Arnold’s book “Huygens and Barrow, Newton and Hooke” (see appendix): https://physicsgg.me/wp-content/uploads/2020/07/huygens-and-barrow-newton-and-hooke-vladimir-i.-arnold.pdf (3) Deriving LRL from the duality: https://arxiv.org/pdf/0803.2610 (2008 on arXiv), https://www.tandfonline.com/doi/pdf/10.1142/S1402925110000726 (2010 on journals) Jakob Hermann’s portrait: https://commons.wikimedia.org/wiki/File:Jakob_Hermann.png Laplace’s portrait: https://commons.wikimedia.org/wiki/File:Laplace,_Pierre-Simon,_marquis_de.jpg Hamilton’s portrait: https://en.wikipedia.org/wiki/File:William_Rowan_Hamilton_portrait_oval_combined.png Gibb’s portrait: https://commons.wikimedia.org/wiki/File:Josiah_Willard_Gibbs_-from_MMS-.jpg Video chapters: 00:00 Introduction 02:08 Maupertuis's principle 09:21 Back to Newtonian gravity 16:46 A "complex" energy 21:43 Complex form vs Vector form 26:44 LRL vector as eccentricity 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!