Ciro Santilli @cirosantilli  40

Functional Updated 2025-07-16

A function that takes input function and outputs a real number.

Euler-Lagrange equation Updated 2025-07-16

Let's start with the one dimensional case. Let the

q : R \to R

and a Functional

I

defined by a function of three variables

F : R^{3} \to R

I (q) = \int_{t_{0}}^{t} F (t, q (t), \overset{q}{˙} (t)) d t

(1)

Then, the Euler-Lagrange equation gives the maxima and minima of the that type of functional. Note that this type of functional is just one very specific type of functional amongst all possible functionals that one might come up with. However, it turns out to be enough to do most of physics, so we are happy with with it.

Given

F

, the Euler-Lagrange equations are a system of ordinary differential equations constructed from that

F

such that the solutions to that system are the maxima/minima.

In the one dimensional case, the system has a single ordinary differential equation:

\frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q} - \frac{d}{d t} \frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q ˙} = 0

(2)

\frac{\partial F}{\partial q}

and

\frac{\partial F}{\partial q ˙}

we simply mean "the partial derivative of

F

with respect to its second and third arguments". The notation is a bit confusing at first, but that's all it means.

Therefore, that expression ends up being at most a second order ordinary differential equation where

q

is the unknown, since:

the term $\frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q}$ is a function of $t, q (t), \overset{q}{˙} (t)$
the term $\frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q ˙}$ is a function of $t, q (t), \overset{q}{˙} (t)$ . And so it's derivative with respect to time will contain only up to $\overset{q}{¨}$

Now let's think about the multi-dimensional case. Instead of having

q : R \to R

, we now have

q : R \to R^{n}

. Think about the Lagrangian mechanics motivation of a double pendulum where for a given time we have two angles.

Let's do the 2-dimensional case then. In that case,

F

is going to be a function of 5 variables

F : R^{5} \to R

rather than 3 as in the one dimensional case, and the functional looks like:

I (q) = \int_{t_{0}}^{t} F (t, q_{1} (t), q_{2} (t), \overset{q_{1}}{˙} (t), \overset{q_{2}}{˙}) (t) d t

(3)

This time, the Euler-Lagrange equations are going to be a system of two ordinary differential equations on two unknown functions

q_{1}

and

q_{2}

of order up to 2 in both variables:

\frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t ))}{\partial q _{1}} - \frac{d}{d t} \frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t )}{\partial q _{1} ˙} = 0 \frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t ))}{\partial q _{2}} - \frac{d}{d t} \frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t )}{\partial q _{2} ˙} = 0

(4)

At this point, notation is getting a bit clunky, so people will often condense the

q_{i}

vector

\frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q _{1}} - \frac{d}{d t} \frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q _{1} ˙} = 0 \frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q _{2}} - \frac{d}{d t} \frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q _{2} ˙} = 0

(5)

or just omit the arguments of

F

entirely:

\frac{\partial F}{\partial q _{i}} - \frac{d}{d t} \frac{\partial F}{\partial q _{i} ˙} = 0

(6)

Video 1.

Calculus of Variations ft. Flammable Maths by vcubingx (2020)

Source.

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Calcium carbonate polymorph Updated 2025-07-16

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ANSI C Updated 2025-07-16

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A closed standard: stackoverflow.com/questions/81656/where-do-i-find-the-current-c-or-c-standard-documents. Nice.

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FreeBSD Updated 2025-07-16

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Atomic Weapons Establishment Updated 2025-07-16

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Their website, and in particular the recruitment section, are so creepy.

There's not mention of bombs. No photos of atomic explosions. The words "atomic" and "weapon" do not even show up in the front page!!! The acronym AWE is instead used everywhere as an euphemism.

In the recruitment section we can see a bunch of people smiling: web.archive.org/web/20211007213222/https://www.awe.co.uk/careers/working-at-awe/, suggesting:

We make nukes, and we do it with a smile!

There's even children outreach!!!

Ciro Santilli is not against storing a few nukes to be ready against dictatorships. But don't be such a pussy! Just say what the fuck you are doing more clearly! You are making weapons to kill people and destroy things in order to maintain the Balance of power. If the public can't handle such facts, then shut down the fucking program.

Video 1.

Keeping the peace: A history of AWE

. Source. By AWE, 2005.

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Nature (journal) Updated 2025-07-16

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Ended up under Springer in 2015 after a massive merger.

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Planck's law Updated 2025-07-16

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Used to explain the black-body radiation experiment.

Published as: On the Theory of the Energy Distribution Law of the Normal Spectrum by Max Planck (1900).

The Quantum Story by Jim Baggott (2011) page 9 mentions that Planck apparently immediately recognized that Planck constant was a new fundamental physical constant, and could have potential applications in the definition of the system of units (TODO where was that published):

Planck wrote that the constants offered: 'the possibility of establishing units of length, mass, time and temperature which are independent of speciﬁc bodies or materials and which necessarily maintain their meaning for all time and for all civilizations, even those which are extraterrestrial and nonhuman, constants which therefore can be called "fundamental physical units of measurement".'

This was a visionary insight, and was finally realized in the 2019 redefinition of the SI base units.

TODO how can it be derived from theoretical principles alone? There is one derivation at; en.wikipedia.org/wiki/Planck%27s_law#Derivation but it does not seem to mention the Schrödinger equation at all.

Video 1.

Quantum Mechanics 2 - Photons by ViaScience (2012)

Source. Contains a good explanation of how discretization + energy increases with frequency explains the black-body radiation experiment curve: you need more and more energy for small wavelengths, each time higher above the average energy available.

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Black-body radiation experiment Updated 2025-07-16

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The Quantum Story by Jim Baggott (2011) page 10 mentions:
Early examples of such cavities included rather expensive closed cylinders made from porcelain and platinum.
and the footnote comments:
The study of cavity radiation was not just about establishing theoretical principles, however. It was also of interest to the German Bureau of Standards as a reference for rating electric lamps.
1859-60 Gustav Kirchhoff demonstrated that the ratio of emitted to absorbed energy depends only on the frequency of the radiation and the temperature inside the cavity
1896 Wien approximation seems to explain existing curves well
1900 expriments by Otto Lummer and Ernst Pringsheim show Wien approximation is bad for lower frequencies
1900-10-07 Heinrich Rubens visits Planck in Planck's villa in the Berlin suburb of Grünewald and informs him about new experimental he and Ferdinand Kurlbaum obtained, still showing that Wien approximation is bad
1900 Planck's law matches Lummer and Pringsheim's experiments well. Planck forced to make the "desperate" postulate that energy is exchanged in quantized lumps. Not clear that light itself is quantized however, he thinks it might be something to do with allowed vibration modes of the atoms of the cavity rather.
1900 Rayleigh-Jeans law derived from classical first principles matches Planck's law for low frequencies, but diverges at higher frequencies.

Video 1.

Black-body Radiation Experiment by sciencesolution (2008)

Source. A modern version of the experiment with a PASCO scientific EX-9920 setup.

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Ultraviolet catastrophe Updated 2025-07-16

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Video 1.

What is the Ultraviolet Catastrophe? by Physics Explained (2020)

Source.

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Bitcoin non-standard transaction Updated 2025-07-16

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Full list at: github.com/cirosantilli/bitcoin-inscription-indexer/blob/master/data/utxo_nonstandard

Bibliography:

Monday, January 29, 2024

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Base58 Updated 2025-07-16

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en.bitcoin.it/wiki/Base58Check_encoding

Invented for Bitcoin.

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Base64 Updated 2025-07-16

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yEnc Updated 2025-07-16

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Big O notation Updated 2025-07-16

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Module bound above, possibly multiplied by a constant:

f (x) = O (g (x))

(1)

is defined as:

\exists M > 0\exists x_{0} \forall x > x_{0} : ∣ f (x) ∣ \leq M g (x)

(2)

E.g.:

$\forall c \in R x + c = O (x)$ . For $c < 0$ , $M = 1$ is enough. Otherwise, any $M > 1$ will do, the bottom line will always catch up to the top one eventually.