I finally find least action principle satisfying

Second channel video: https://youtu.be/FWjl5ZuBs3s The action principle has always been deeply unsatisfying to me. Why cook up this quantity? Why should we minimise it? I finally found a satisfying answer, which is made more satisfying by Noether's theorem. 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: actionistautological Sources: This video is basically an amalgamation of the paper: https://www.nature.com/articles/s41598-023-39145-y (with more details here: https://static-content.springer.com/esm/art%3A10.1038%2Fs41598-023-39145-y/MediaObjects/41598_2023_39145_MOESM1_ESM.pdf) and the Wikipedia article on Noether’s theorem: https://en.wikipedia.org/wiki/Noether%27s_theorem#Brief_illustration_and_overview_of_the_concept However, I do have to warn that the paper uses pretty unconventional notations, which were confusing to me at first, so I have changed the notations considerably. Rabbit holes to dive into: Ostrogradsky instability: https://en.wikipedia.org/wiki/Ostrogradsky_instability https://arxiv.org/pdf/1209.0583 https://www.ctc.cam.ac.uk/activities/cosmo2013/presentations/Tai-junChen.pdf (a presentation of the former preprint) https://arxiv.org/pdf/2403.19777 Cleverly choosing coordinates to make vector field divergenceless: https://hal.science/hal-04215419/document (Are there similar works to this? Please let me know.) https://physics.stackexchange.com/questions/432664/is-every-autonomous-first-order-planar-2d-system-integrable/432672#432672 (Seems like this is quite a general argument for a 2D system - I don’t have access to the book referenced, so maybe there are more details for higher-dimensional systems) Hamilton-Jacobi equations: https://arxiv.org/pdf/1910.09414 https://www.math.wustl.edu/~alanchang/files/alan_chang_junior_paper_hamilton_jacobi_equations.pdf Video chapters: 00:00 Intro 00:41 Alternative definition of action 04:36 Why action is integral of T - V 07:07 Gauge dependence of action 12:00 Why minimise S? 16:23 Noether's theorem 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!