#PhysicsFactlet
One more thread has found its way to my website. This time is the turn of variational calculus!
https://jacopobertolotti.com/VariationalCalculusIntro.html

#PhysicsFactlet
In the past I have made a number of threads on Xwitter giving brief explanations of bits of Physics. I now need a less volatile place where to keep them, so I started to copy-paste them into my personal page. I assume it will take me forever to transfer even just a small part of them, but here is a starting point: https://jacopobertolotti.com/LagrangianIntro.html

#PhysicsFactlet
Does a chaotic system always behave chaotically?
Not really, as many chaotic systems have a subset of possible initial conditions that lead to a quasi-periodic motion.
As an example, below are two sets (black and orange) of 20 double pendula each, all with the same initial energy, and each group starting with very similar initial conditions.
The first group (black) spread out a little bit with time, but nearby initial conditions keeps evolving into nearby dynamics, which is typical of integrable systems.
On the other hand the pendula in the second group (orange) also starts with similar initial condition, but after a short transient evolve each very differently from each other, which is a mark of a chaotic system.
#Physics #Visualization #Chaos

#PhysicsFactlet
Can shadows move faster than light?
Not really. There is nothing moving sideways, so nothing is moving faster than light (which, incidentally, mean you can't use shadows to communicate faster than light).
But the edges of the projection of the shadow can indeed appear to move arbitrarily fast.
#ITeachPhysics #Visualization #Physics #Optics

#PhysicsFactlet
The van Cittert–Zernike theorem is usually phrased in terms of fringe visibility, but a simpler way to look at it is that a incoherent source seen from far away enough will look like a point source (i.e. spatially coherent).

Explanation of the plot:
The three sources are incoherent with each other.
The grey lines are the zeros of the field from each source (at a fixed time), and the black line the zero of the total field.
When we move further away those lines look more and more like straight line.

#Physics #Optics #Visualization

[Not really a #PhysicsFactlet]
And while the whole Astro community is super excited about pulsars and gravitational waves, a very nice result from my corner of Physics that will likely go unnoticed by most.
Let's talk Anderson Localization!

(If you are not familiar with what Anderson localization is, I made a beginner-friendly intro on the birdsite some time back: https://twitter.com/j_bertolotti/status/1525033971670781953?s=20)

Thread: 1/
#Physics #Optics

#PhysicsFactlet
Magnetic hysteresis: In a ferromagnet the equilibrium configuration is with all magnetic moments aligned with each other. If we want to flip them, we need to flip all of them at the same time, which requires a stronger field than if the moments were independent, resulting in the characteristic hysteresis loop.

(Simulation done by numerically solve the Landau–Lifshitz equation with a tiny bit of noise added to speed the process up on a square grid of magnetic moment with periodic boundary conditions.)

#Physics #Visualization #ITeachPhysics #Magnetism

#PhysicsFactlet
"Anderson localization" is a weird phenomenon that is not well known even among Physicists, but has the habit of popping up essentially everywhere.
So here is an introductory thread 🧵

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. 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.
The original paper (and, frankly, most of the literature on the subject) is pretty impenetrable, but thankfully Anderson localization can happen any time we have a wave and a random medium, doesn't matter what kind of wave, so we can try to look at a simple system.

Let's start VERY simple with the simple pendulum, and let's make it even simpler by assuming the oscillations are small, and thus we only have to deal with an harmonic oscillator. The pendulum has a natural frequency (i.e. the frequency at which it will naturally oscillate if you just let it go), which will depend on its length and the gravity acceleration pulling it down: ω₀= √(g/L). If you take a bunch of such pendula, they will all oscillate with the same natural frequency.
Let's complicate the problem a bit and add a (elastic) coupling between the pendula. The system in its entirety now will have a number of natural frequencies equal to the number of pendula, resulting in a complex motion that is the superposition of the oscillations at all those different frequencies.
1/

#PhysicsFactlet
At optical frequencies, most media respond to an electromagnetic wave proportionally to the electric (magnetic) field applied, and are thus said to be "linear".
In such media a monochromatic wave can be slowed down, resulting in refraction, but its frequency is unchanged.
But for all real media, the reaction to an electromagnetic field is only approximately linear, and for high fields it is possible to see deviation from linearity.
One consequence of a nonlinear response, is that the medium can generate frequencies that were not present in the pump wave, the simplest case being the "second harmonic generation" where a wave of twice the frequency of the pump is produced.
But, due to the time reversal symmetry of Maxwell equations, if we can generate a wave at twice the frequency, we can also take that and convert it back to the original frequency.
So, in order to maximise the amount of second harmonic generated, one needs to make sure that the phase of the pump and the second harmonic do not drift while the waves are propagating through the crystal. If these "phase matching conditions" are not satisfied, the energy will flow back from the second harmonic to the pump.

#Physics #Optics #SciComm #SciViz #Visualization

#PhysicsFactlet
In a dielectric medium, the charges are not free to move around, so electromagnetic waves are slowed down a bit, but otherwise can freely propagate.
In a metal, some of the electrons can move around, and thus tend to absorb and/or reflect electromagnetic waves.
But at the interface between a dielectric and a metal it is possible to have a mode that is part electron oscillation and part electromagnetic field, travelling along the interface, known as a "surface plasma polariton" (sometimes abbreviated as "plasmon").

Shown in the animation is the electric field (which is a vector quantity, hence the arrows).

#Physics #Optics #Plasmonics #Visualization #SciComm

#PhysicsFactlet
(Made for a student) For the single frequency oscillation of an infinite chain of identical masses connected by identical springs, the relative phase of oscillation of adjacent masses will depend non-linearly with frequency, resulting in travelling or stationary waves.
For the "diatomic chain", where two different masses alternate, the dependence of relative phase with frequency has two branches (separated by a gap of not allowed frequencies).
#Physics #CondensedMatter #Visualization

#PhysicsFactlet
A way to visualize the transition to chaos of a mechanical system is to look how its Poincaré section(s) changes when the perturbation increases.
In this example a damped simple pendulum with a sinusoidal forcing of amplitude "f", starting at rest.
For small forcing, the motion is periodic, so the Poincaré section is just a single point. As the forcing increases we observe multiple bifurcations, aperiodic motion, a second regime of periodic motion, and then the Poincaré section becomes a fractal curve.
#Chaos #Physics #Visualization

RT @j_bertolotti
#PhysicsFactlet #MadeForALecture
Even for a chaotic system (in this case a kicked rotor), not all initial conditions are going to show a chaotic dynamics. Near a hyperbolic fixed point (purple points) one will get chaos, but not near an elliptic one (orange points).

RT @j_bertolotti
#PhysicsFactlet #MadeForALecture
Even for a chaotic system (in this case a kicked rotor), not all initial conditions are going to show a chaotic dynamics. Near a hyperbolic fixed point (purple points) one will get chaos, but not near an elliptic one (orange points).

#PhysicsFactlet #MadeForALecture
Even for a chaotic system (in this case a kicked rotor), not all initial conditions are going to show a chaotic dynamics. Initial conditions near an elliptic fixed point (orange) have a "nice" pendulum-like dynamics. But initial conditions starting next to a hyperbolic fixed point (purple) go on a trajectory that gets more and more oscillating, leading to chaos.
#Physics #Visualization #Chaos

#PhysicsFactlet #WorkinProgress
Need to explain students about fixed points in dynamical systems, and I am making a plot of the most common cases (still not sure whether animating it would make it more or less understandable).
#Visualization #Physics

#PhysicsFactlet
A light pulse can be thought as the superposition of a large number of sinusoidals with different frequencies.
In vacuum, all frequencies travel at the same speed, and so the pulse remains identical to itself while propagating.
But in a medium (e.g. air, water, glass, etc) different frequencies travel at different speeds, a phenomenon known as "dispersion", and thus the pulse stretches while propagating.
#Physics #Optics # Visualization

#PhysicsFactlet
The dynamics of a rigid sphere can be surprisingly complicated. In fact, beside rotating around an axis, the sphere can also precess (i.e. the axis of rotation is itself rotating) and nutate (i.e. the axis of rotation oscillates back and forth).

Euler's angles are particularly useful to describe such motion, as:
* Rotation is a change in the third Euler angle.
* Precession is a change in the first Euler angle.
* Nutation is a change in the second Euler angle.

#Physics #ClassicalMechanics #EulerAngles #Visualization

#PhysicsFactlet
The velocity of propagation of a sea wave depends on the depth of the water: the shallower the water the slower the wave. So if the depth reduces the wave will "accumulate" (like in a traffic jam).
If the liquid is incompressible, this can create a Tsunami.

Disclaimer: Just solving the wave equation with the appropriate dispersion relation here, to show the "traffic jam" effect. NOT a proper fluid dynamics simulation.
#Physics #Visualization #Tsunami #GravityWaves

#PhysicsFactlet
There is often a lot lost in translation when talking across scientific fields. One often misunderstood point is why in optics/photonics we never use p-values, while they are everywhere in biology&co.
The point is that in optics we usually have a ton of data, so even a very poor signal (e.g. signal to noise ~1) has a miniscule p-value, making it not very useful as a metric.
#Statistics #Optics #Photonics #Physics #Visualization