/sci/ - Science & Math

>>12518086
be Turing complete?

>>12518097
The Lagrangian here is a map
[math]L\colon {\mathbb R}^d\times {\mathbb R}^d\times {\mathbb R} \to {\mathbb R} [/math],
i.e. "a function [math]L(x,v,t)[/math]"
The derivative "[math]\frac{\partial L}{\partial q'(t)}[/math]" is [math]\left(\frac{\partial L}{\partial v}\right)_{v=q'(t)}[/math], so you don't need to worry about correlations between [math]q'(t)[/math] and [math]q(t)[/math] here.

There's the ubiquitous habit of also denoting then t-parametrized value "[math]L(q(t), q'(t), t)[/math]", a function
[math]{\mathbb R} \to {\mathbb R} [/math]
by [math]L[/math], so don't get confused.

Also don't mix up partial derivatives, as in [math]\left(\frac{\partial L}{\partial v}\right)_{v=q'(t)}[/math], with total ones, as in [math]\frac{d}{dt}\left(\frac{\partial L}{\partial x}\right)_{x=q(t)}[/math]

Lastly, there's variations of functionals [math]\frac{\delta S}{\delta q}[/math], where e.g. [math]S\colon ({\mathbb R}\to {\mathbb R}^d)\to {\mathbb R}[/math], so don't confuse those either.
If you read a good of physics textbooks, you'll get very confused with d'Alembert's principle (and if you watch random undergrads on youtube you'll get away with the idea that it's just a trivial rewrite of Newton, so I'm already sceptical). If you read a math textbook you'll have some work bringing those together with the physics books but if you look into the post Arnold slash control theory slash theoretical engineering, say, diff geo formulations that make it rigorous, you get

https://en.wikipedia.org/wiki/Virtual_displacement#Definition

so just learn the rules and don't mix up [math]d, \partial, \delta[/math].

And with that said, watch out for L's that are neither maps on [math](x,v,t)[/math] nor [math]t[/math] but actually functionals eating [math]t\mapsto x[/math].

(So [math]\frac{\delta q(t)}{\delta q'(t)}[/math] is not that you want to compute, but you could probably give meanint to it via some diract delta functionals)

>>12424046
In physics, you see those concepts e.g. in field theory
https://en.wikipedia.org/wiki/Grassmann_number

or general more wilder representations of symmetry groups (and half-integer spin fields).
But it's also a representation of a special case of a quotient of a polynomial ring and common in algebraic geometry, when more differential geometry-like properties want to be modeled.

I skimmed through the Wildberger video and I'm super surprised he talks positively of Robinson's non-standard analysis. Not because I dislike it, but - as he points out - it's not much less "problematic" than the reals. Seems he just want to be a contrarian. Non-standard analysis periodically has its comeback and it's logic formulation is, I think, over considered to be too complicated (given people are already used to standard foundations for math).

So I know what we'll see now - him working out how to phrase general theorems of the theory in its linear representation (and his two "colored complex units"). I'm find with this, but his non-formalist othodoxy is tiresome and I call into question his self-proclaimed rogue status

Because we're a bunch of pretentious elitists.

>>2831780
You say the Lorentz group (ISO(1,3) if you will) seems to have no deeper meaning than electromagnetism/maxwell equations and your argument is that the later can be described using a U(1) fibre.

Well, you can also describe any holonomy group (like the lorentz group) using a fibre (Hauptfasenbündel in this case) - so how does your argument still hold?

The last sentence was just the statement "you could imagine any fibre to be present in nature, not just
>its the possible symmetry groups to U(1)_Y\otimes SU(2)_L\otimes SU(3)_C\otimes\mathrm{GravityWhateverGroup}
so why has
>U(1)
a deeper meaning than
>\mathrm{GravityWhateverGroup}

okay, but this can't be the original pic. can't believe the artist would draw all these people and then use word to write shitty styled formulas

Would we survive if earth was 10 times bigger than its current size or is it just perfect as it is ?