Discover and read the best of Twitter Threads about #PhysicsFactlet
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#PhysicsFactlet
An attempt to explain what tensors are for people with high-school Math (if you are a mathematician, this thread is not for you).
Not sure why, but tensors are often introduced in a very confused way, that makes them look more scary than they actually are.
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An attempt to explain what tensors are for people with high-school Math (if you are a mathematician, this thread is not for you).
Not sure why, but tensors are often introduced in a very confused way, that makes them look more scary than they actually are.
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Let's assume you are familiar with matrices (if you aren't, chances are you don't care what a tensor is), so the fact that multiplying rows by columns a row vector with a column vector yields a scalar (i.e. a single number) should be no surprise to you.
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If we make a column of row vectors, we can repeat the process for each of them and put the results also in a column, resulting in the usual multiplication of a matrix by a vector.
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#PhysicsFactlet (342) Lagrange multipliers
Strictly speaking Lagrange multipliers are not "Physics", but they are so useful to solve so many Physical problems, that it is definitively worth looking at them.
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Strictly speaking Lagrange multipliers are not "Physics", but they are so useful to solve so many Physical problems, that it is definitively worth looking at them.
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Before we even introduce them, let's solve a super-simple problem, which will form the basis for our motivation to look into Lagrange multipliers:
Find the minimum of the function f=x²+y².
Yes, I can hear you shouting x=y=0, but let's still do the calculation.
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Find the minimum of the function f=x²+y².
Yes, I can hear you shouting x=y=0, but let's still do the calculation.
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The way you find the minimum of a function is to check the points where all the partial derivatives are zero (in this case we have 2 variables, so we will look at the partial derivatives with respect to x and y): df/dx=2 x, df/dy=2y --> 2x=0, 2y=0 --> x=y=0.
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#PhysicsFactlet (335)
Yesterday, at a small playground where my son was playing, I saw this Kugel fountain, so here comes a short thread about Kugel fountains and how they work.
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(Alt Text: a Kugel fountain slowly rotating in a sunny day.)
Yesterday, at a small playground where my son was playing, I saw this Kugel fountain, so here comes a short thread about Kugel fountains and how they work.
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(Alt Text: a Kugel fountain slowly rotating in a sunny day.)
First of all, what is a Kugel fountain?
There are a few variations on the theme, but usually they are big stone spheres, sitting on a hemispherical hole, with water flowing from below. Despite their weight, they can spin with a small push, and keep spinning for a long time.
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There are a few variations on the theme, but usually they are big stone spheres, sitting on a hemispherical hole, with water flowing from below. Despite their weight, they can spin with a small push, and keep spinning for a long time.
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How does it work?
It can't be buoyancy, as the stone sphere is a a LOT more dense than the water (we all have direct experience of stones sinking when you put them in water, and this one is not any different).
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It can't be buoyancy, as the stone sphere is a a LOT more dense than the water (we all have direct experience of stones sinking when you put them in water, and this one is not any different).
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#PhysicsFactlet (331)
"Anderson localization" is a weird phenomenon that is not well known even among Physicists, but has the habit of popping up essentially everywhere.
An introductory thread 🧵
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"Anderson localization" is a weird phenomenon that is not well known even among Physicists, but has the habit of popping up essentially everywhere.
An introductory thread 🧵
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The idea of "localization" originally came about as an explanation (by P.W. Anderson, hence the name) of why the spins in certain materials did not relax as fast as expected.
nobelprize.org/prizes/physics…
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nobelprize.org/prizes/physics…
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What Anderson realized was that when you have a wave (in this case a quantum mechanical wavefunction) that propagates in a random system, interference can play a major role, and potentially impede propagation completely.
journals.aps.org/pr/abstract/10…
(Paywalled)
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journals.aps.org/pr/abstract/10…
(Paywalled)
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#PhysicsFactlet The ones I am most proud of from 2021 (in chronological order)
A visualization of what an eigenvector is (at least for 2x2 matrices)
A visualization of what an eigenvector is (at least for 2x2 matrices)
Pulse chirping (keeping the pulse duration constant for ease of visualization, although in reality one usually keeps the bandwidth constant)
A simplified picture of spin waves.
#PhysicsFactlet (308)
There are not many problems in Physics that can be solved exactly, so we tend to rely on perturbation theory a lot. One of the problems with perturbation theory is that infinities have the bad habit of popping up everywhere when you use it.
(A thread 1/ )
There are not many problems in Physics that can be solved exactly, so we tend to rely on perturbation theory a lot. One of the problems with perturbation theory is that infinities have the bad habit of popping up everywhere when you use it.
(A thread 1/ )
If you know anything about Physics you are probably thinking about quantum field theory and all the nasty infinities that we need to "renormalize". But quantum field theory is difficult, so let's look at a MUCH simpler problem: the anharmonic oscillator.
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Disclaimer: I can't know how much you (the reader) know about this. For some of you this thread will be full of obvious stuff. For others there will be so many missing steps to be hard to follow. I will do my best, but I apologize with everyone in advance.
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#PhysicsFactlet (299)
Fractional derivatives: a brief tutorial/🧵. If you know some calculus you should be able to follow. If you are a Mathematician (or you like to see things done properly) I advise "Fractional Differential Equations" by I. Podlubny instead 😉
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Fractional derivatives: a brief tutorial/🧵. If you know some calculus you should be able to follow. If you are a Mathematician (or you like to see things done properly) I advise "Fractional Differential Equations" by I. Podlubny instead 😉
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The history of fractional derivatives begins together with the history of the much more common integer-order derivatives, and a number of big names in mathematics worked on it over the centuries.
Afaik, the first to work on the problem was Leibniz himself.
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Afaik, the first to work on the problem was Leibniz himself.
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Since differentiating twice a function yields the second derivative, the Marquis de l'Hôpital immediately wondered whether it makes sense to think about an operator which, if applied twice, gave the first derivative, i.e. some sort of derivative of order 0.5
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#PhysicsFactlet (283)
Lorentz transformations pre-date Special Relativity. How is that even possible?
A thread.
Trigger warning for typos (hopefully just in the text and not in the equations) and carefree manipulations of equations 😉
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Lorentz transformations pre-date Special Relativity. How is that even possible?
A thread.
Trigger warning for typos (hopefully just in the text and not in the equations) and carefree manipulations of equations 😉
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The historical route is interesting but complicated, so I will leave that story for someone more qualified to write it. What I want to look at is: how do we get the Lorentz transformations without knowing anything about special relativity?
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A requirement we want all physical theories to satisfy is the "principle of relativity", i.e. the fact that the laws of Physics are the same in every frame of reference. Were this not the case, each of us would experience a different universe, making life quite complicated.
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#PhysicsFactlet (273)
A brief introduction to the calculus of variations.
Trigger warning: lots of formulas manipulated the way experimental physicists do 🙂
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A brief introduction to the calculus of variations.
Trigger warning: lots of formulas manipulated the way experimental physicists do 🙂
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The simplest introduction to the calculus of variations is to solve in a slightly roundabout way a very easy geometrical problem: what is the shortest path between 2 points on a plane?
(spoiler: it's a straight line)
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(spoiler: it's a straight line)
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Let's pretend we have no idea, and so we are forced to take into consideration all possible functions passing through 2 given points. What we want to do is to calculate the length of each of them, and select the shortest one.
(spoiler: we are not REALLY going to do that)
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(spoiler: we are not REALLY going to do that)
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Moving all the old Physics factlects to the same hashtag to make easier to find them. As Twitter does not allow me to edit my tweets, I need to repost all of them. Apologies if this floods your timeline.
#Physicsfactlet (1)
The uncertainty principle is not a principle, it is a theorem. Just like the Pauli exclusion principle and many others. It was a principle when it was first formulated, but we have since realised that it can be derived from first principles.
The uncertainty principle is not a principle, it is a theorem. Just like the Pauli exclusion principle and many others. It was a principle when it was first formulated, but we have since realised that it can be derived from first principles.
#Physicsfactlet (2)
There is nothing quantum in describing particles as waves. In fact one can rewrite the whole Newtonian mechanics as a wave theory without changing any result or prediction. (See Hamilton-Jacobi formalism)
There is nothing quantum in describing particles as waves. In fact one can rewrite the whole Newtonian mechanics as a wave theory without changing any result or prediction. (See Hamilton-Jacobi formalism)
