Ciro Santilli @cirosantilli 37

for topological manifolds: this is a generalization of the Poincaré conjecture.
Original problem posed, $n = 3$ for topological manifolds.
Millennium Prize Problems.
Last to be proven, only the 4-differential manifold case missing as of 2013.
Even the truth for all $n > 4$ was proven in the 60's!
Why is low dimension harder than high dimension?? Surprise!
AKA: classification of compact 3-manifolds. The result turned out to be even simpler than compact 2-manifolds: there is only one, and it is equal to the 3-sphere.
For dimension two, we know there are infinitely many: classification of closed surfaces
for differential manifolds:
Not true in general. First counter example is $n = 7$ . Surprise: what is special about the number 7!?
Counter examples are called exotic spheres.
Totally unpredictable count table:
Dimension Smooth types
1 1
2 1
3 1
4 ?
5 1
6 1
7 28
8 2
9 8
10 6
11 992
12 1
13 3
14 2
15 16256
16 2
17 16
18 16
19 523264
20 24
$n = 4$ is an open problem, there could even be infinitely many. Again, why are things more complicated in lower dimensions??

Dimension	Smooth types
1	1
2	1
3	1
4	?
5	1
6	1
7	28
8	2
9	8
10	6
11	992
12	1
13	3
14	2
15	16256
16	2
17	16
18	16
19	523264
20	24

 Read the full article

Genius  Updated 2025-05-23  +Created 1970-01-01

 View more

This is not a label that Ciro Santilli likes to give lightly. But maybe sometimes, it is inevitable.

Bibliography:

www.geniuses.club

 Read the full article

Get nucleotide at a given position of a FASTA file  Updated 2025-05-23  +Created 1970-01-01

 View more

www.biostars.org/p/263478/

 Read the full article

Glueball  Updated 2025-05-23  +Created 1970-01-01

 Read the full article

Lagrangian mechanics  Updated 2025-05-23  +Created 1970-01-01

 View more

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:

compound Atwood machine. Here, we can use the coordinates as the heights of masses relative to the axles rather than absolute heights relative to the ground
double pendulum, using two angles. The Lagrangian approach is simpler than using Newton's laws
- pendulum, use angle instead of x/y
two-body problem, use the distance between the bodies

lagrangian mechanics lectures by Michel van Biezen (2017) is a good starting point.

When doing lagrangian mechanics, we just lump together all generalized coordinates into a single vector

q (t) : R \to R^{n}

that maps time to the full state:

q (t) = [q_{1} (t), q_{2} (t), q_{3} (t), ..., q_{n} (t)]

(1)

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

R^{n} \times R^{n} \times R \to R

that maps:

L (q (t), \dot{q} (t), t)

(2)

to a real number.

Then, the stationary action principle says that the actual path taken obeys the Euler-Lagrange equation:

\frac{\partial L ( q ( t ) , q ˙ ( t ) , t )}{\partial q _{i}} - \frac{d}{d t} \frac{\partial L ( q ( t ) , q ˙ ( t ) , t )}{\partial q _{i} ˙} = 0

(3)

This produces a system of partial differential equations with:

$n$ equations
$n$ unknown functions $q_{i}$
at most second order derivatives of $q_{i}$ . Those appear because of the chain rule on the second term.

The mixture of so many derivatives is a bit mind mending, so we can clarify them a bit further. At:

\frac{\partial L ( q ( t ) , q ˙ ( t ) , t )}{\partial q _{i}}

(4)

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:

\frac{d}{d t} \frac{\partial L ( q ( t ) , q ˙ ( t ) , t )}{\partial q _{i} ˙}

(5)

the $\frac{\partial}{\partial q _{i} ˙}$ part is just like the previous term, $\overset{q_{i}}{˙}$ just identifies the argument with index $n + i$ ( $n$ because we have the $n$ non derivative arguments)
after the partial derivative is taken and returns a new function $\frac{\partial L ( q ( t ) , q ˙ ( t ) , t )}{\partial q _{i} ˙}$ , then the multivariable chain rule comes in and expands everything into $2 n + 1$ terms

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: S.

TODO advantages:

Bibliography:

www.physics.usu.edu/torre/6010_Fall_2010/Lectures.html Physics 6010 Classical Mechanics lecture notes by Charles Torre from Utah State University published on 2010,
- Classical physics only. The last lecture: www.physics.usu.edu/torre/6010_Fall_2010/Lectures/12.pdf mentions Lie algebra more or less briefly.
www.damtp.cam.ac.uk/user/tong/dynamics/two.pdf by David Tong

Video 1.

Euler-Lagrange equation explained intuitively - Lagrangian Mechanics by Physics Videos by Eugene Khutoryansky (2018)

Source. Well, unsurprisingly, it is exactly what you can expect from an Eugene Khutoryansky video.

 Read the full article

List of search engines  Updated 2025-05-23  +Created 1970-01-01

 Read the full article

Porn meme  Updated 2025-05-23  +Created 1970-01-01

 Read the full article

 There are unlisted articles, also show them or only show them.