Ciro Santilli @cirosantilli 35

These people don't fuck around.

Basically a social network where you don't know the other people very well.

Originally it was likely created to study constrained mechanical systems where you want to use some "custom convenient" variables to parametrize things instead of global x, y, z. Classical examples that you must have in mind include:

When doing lagrangian mechanics, we just lump together all generalized coordinates into a single vector $\overrightarrow{q}(t):\mathbb{R}\to {\mathbb{R}}^n$ that maps time to the full state:where each component can be anything, either the x/y/z coordinates relative to the ground of different particles, or angles, or nay other crazy thing we want.

The Lagrangian is a function ${\mathbb{R}}^n\times {\mathbb{R}}^n\times \mathbb{R}\to \mathbb{R}$ that maps:to a real number.

Then, the stationary action principle says that the actual path taken obeys the Euler-Lagrange equation:This produces a system of partial differential equations with:

The mixture of so many derivatives is a bit mind mending, so we can clarify them a bit further. At:the $q_i$ is just identifying which argument of the Lagrangian we are differentiating by: the i-th according to the order of our definition of the Lagrangian. It is not the actual function, just a mnemonic.

Then at:

However, people later noticed that the Lagrangian had some nice properties related to Lie group continuous symmetries.

Basically it seems that the easiest way to come up with new quantum field theory models is to first find the Lagrangian, and then derive the equations of motion from them.

For every continuous symmetry in the system (modelled by a Lie group), there is a corresponding conservation law: local symmetries of the Lagrangian imply conserved currents.
Genius: Richard Feynman and Modern Physics by James Gleick (1994) chapter "The Best Path" mentions that Richard Feynman didn't like the Lagrangian mechanics approach when he started university at MIT, because he felt it was too magical. The reason is that the Lagrangian approach basically starts from the principle that "nature minimizes the action across time globally". This implies that things that will happen in the future are also taken into consideration when deciding what has to happen before them! Much like the lifeguard in the lifegard problem making global decisions about the future. However, chapter "Least Action in Quantum Mechanics" comments that Feynman later notice that this was indeed necessary while developping Wheeler-Feynman absorber theory into quantum electrodynamics, because they felt that it would make more sense to consider things that way while playing with ideas such as positrons are electrons travelling back in time. This is in contrast with Hamiltonian mechanics, where the idea of time moving foward is more directly present, e.g. as in the Schrödinger equation.

Furthermore, given the symmetry, we can calculate the derived conservation law, and vice versa.

And partly due to the above observations, it was noticed that the easiest way to describe the fundamental laws of particle physics and make calculations with them is to first formulate their Lagrangian somehow: why do symmetries such as SU(3), SU(2) and U(1) matter in particle physics?s.

TODO advantages:

Bibliography:

Since a matrix $M$ can be seen as a linear map $f_M(\vec{x})$, the product of two matrices $MN$ can be seen as the composition of two linear maps:One cool thing about linear functions is that we can easily pre-calculate this product only once to obtain a new matrix, and so we don't have to do both multiplications separately each time.

By Evan Chen (陳誼廷)

800+ page PDF with source on GitHub claiming to try and teach the beauty of modern maths for high schoolers. Fantastic project!!!

TODO find a concrete numerical example of doing calculus on a differentiable manifold and visualizing it. Likely start with a boring circle. That would be sweet...

Ciro Santilli is fond of cynicism, see also ciro Santilli's film tastes.

Kudos for being a not-for-profit. Also, anyone can create content: e-learning websites must allow students to create learning content. Oh, but TODO is possible for anyone to make content publicly visible? Course join links lik: www.khanacademy.org/join/MJZ6NSV7 require login. webapps.stackexchange.com/questions/165132/how-to-create-a-course-that-is-publicly-visible-without-the-need-to-login-on-kha If that's the case, it is a fatal flaw not shared by OurBigBook.com.

Another cool aspect is that they have the "physical world teacher pull student accounts in" approach built-in quite well at course creation. This is a very good feature.

As of 2021 they were a bit struggling for money it seems: www.youtube.com/watch?v=I8XdUy-wyyM?

A form is a function from a vector space to elements of the underlying field of the vector space.

Examples: