As mentioned at buzzard.ups.edu/courses/2017spring/projects/schumann-lie-group-ups-434-2017.pdf, what the symmetry (Lie group) acts on (obviously?!) are the Lagrangian generalized coordinates. And from that, we immediately guess that manifolds are going to be important, because the generalized variables of the Lagrangian can trivially be Non-Euclidean geometry, e.g. the pendulum lives on an infinite cylinder.
Equivalent to Lagrangian mechanics but formulated in a different way.
Motivation: Lagrangian vs Hamiltonian.
TODO understand original historical motivation, www.youtube.com/watch?v=SZXHoWwBcDc says it is from optics.
Intuitively, the Hamiltonian is the total energy of the system in terms of arbitrary parameters, a bit like Lagrangian mechanics.
Bibliography:
Analogous to what the Euler-Lagrange equation is to Lagrangian mechanics, Hamilton's equations give the equations of motion from a given input Hamiltonian:So once you have the Hamiltonian, you can write down this system of partial differential equations which can then be numerically solved.
youtu.be/Ca7c5B7Js18?t=803 compares Lagrangian mechanics equation vs the direct x/y coordinate equation.
Morgan Ward refers to multiple subjects, so the context in which you encountered the term would determine its specific meaning. 1. **Morgan Ward as a Name**: It could refer to a person, such as an individual with that name who might be known in various professional fields like art, academia, or business. 2. **Morgan Ward in Geographic Context**: It might also refer to an area or district named Morgan Ward, which could be a division in a city, town, or municipality.
Became very popular as of result of people using Bullet Physics for reinforcement learning AI training robot simulations.
Source code: somewhere inside the main Bullet Physics source tree. Yay.
Bozenna Pasik-Duncan is a mathematician known for her contributions to various fields including mathematics education, differential equations, and dynamical systems. She has been involved in research and has published numerous papers and articles. Additionally, she has been active in promoting mathematics education and has engaged in outreach activities to inspire and educate students in mathematics.
W. Ross Ashby, or William Ross Ashby (1903–1972), was a British cybernetician and psychiatrist known for his contributions to the fields of cybernetics, systems theory, and complexity science.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. Web editor. You can also edit articles on the Web editor without installing anything locally.Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact