Chaos and the Double Pendulum

chaotic system is one in which infinitesimal differences in the starting conditions lead to drastically different results as the system evolves.

Summarized by mathematician Edward Lorenz, ”Chaos [is] when the present determines the future, but the approximate present does not approximately determine the future.”

There’s an important distinction to make between a chaotic system and a random system. Given the starting conditions, a chaotic system is entirely deterministic. A random system, on the other hand, is entirely non-deterministic, even when the starting conditions are known. That is, with enough information, the evolution of a chaotic system is entirely predictable, but in a random system there’s no amount of information that would be enough to predict the system’s evolution.

The simulations above show two slightly different initial conditions for a double pendulum — an example of a chaotic system. In the left animation both pendulums begin horizontally, and in the right animation the red pendulum begins horizontally and the blue is rotated by 0.1 radians (≈ 5.73°) above the positive x-axis. In both simulations, all of the pendulums begin from rest.

Mathematica code posted here.

[For more information on how to solve for the motion of a double pendulum, check out my video here.]

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Illustris Simulation of the Universe
Video Credit: Illustris Collaboration, NASA, PRACE, XSEDE, MIT, Harvard CfA

How did we get here? Click play, sit back, and watch. A new computer simulation of the evolution of the universe — the largest and most sophisticated yet produced — provides new insight into how galaxies formed and new perspectives into humanity’s place in the universe.

Read More

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Signal Collection and Parabolic Reflectors

reflector is a type of antenna that receives and focuses various types of signals. Reflectors have numerous applications, from satellite dishes and telescopes, to long-distance microphones and car headlights. One common feature of these examples is their parabolic shape, giving them the name parabolic reflectors.

It turns out that paraboloids are the perfect shape for focusing signals from distant sources. When pointed directly at the the incoming signal, a parabolic reflector (GIF 1) collects the signal to a single focal point, where a receiver, called a feed horn, is placed to collect the focused transmission.

In many applications, parabolic reflectors are too costly to produce, so spherical reflectors (GIF 2) are used instead. The disadvantage of spherical reflectors is that they have multiple focal points, and therefore produce blurry results.

Mathematica code posted here.

This code is incredibly messy and I guarantee there’s a better way to calculate this. Please contact me if you have suggestions!

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Accretion Disks

Accretion flows are ubiquitous in astrophysics: they occur around protostars, accreting compact objects in binary systems, and supermassive black holes at the cores of galaxies. Much of professor James M. Stone's work has concerned studies of the local hydrodynamic and magnetohydrodynamics (MHD) processes that can lead to outward angular momentum transport in accretion disks. As computers become more powerful, previous studies of local patches of an accretion flow are being expanded into global studies that encompass the entire disk.

Accretion flows that cannot cool via emission of radiation become vertically thick and nearly spherical. Thus, they are intrinsically multidimensional. To study the structure and evolution of non-radiative accretion flows, 2D (axisymmetric) hydrodynamical simulations were performed using a non-uniform grid that spanned more than two decades in radius.

The most striking property of the flow is the large fluctuations produced by strong convection. Convective eddies transport a lot of mass both inwards and outwards, but the net mass accretion rate is very small and set by the properties of the flow near the inner boundary. A vanishingly small accretion rate may help to explain the deficit of high energy emission observed from accreting compact sources.

While understanding the properties of hydrodynamical accretion flows is important, it is generally agreed that angular momentum transport is in fact mediated by magnetic stresses. Thus, repeating the global simulations of non-radiative accretion flows with MHD calculations is vital.

Credit: James M. Stone

Hello everyone from the tags i have chosen to promote this in. I’m here to inform everyone that I have put together a proposal for the NA company NISAmerica,   for the localization of the title “Tongari Boushi to Mahou no Machi.” The 3DS sequel to the game known either as “Magician’s Quest, Mysterious Times” for the US region, or as “Enchanted Folk and the School of Wizardy” for EU Region players.  

Its a game in the vein of Animal Crossing but sets itself apart with features such as the ability to run your own shop, make your own items/accessories and learn magic via a magic school and classes. It even contains the feature to learn and play music with villagers and players, There’s even a Story mode; where you can progress the game’s story once every week via “Mystery Time”.

Other features include A customizable mall, and highly customizable fashion.  The ability to “date” villagers or “students” and using magic to interact with them as well as your town. 

The proposal to NIS America can be found here,

I have also submitted a proposal to XSeed, the group that localized Rune Factory 4!

and you can read more on the details of the game there. If you are a NIS America or XSeed member, any support showing your interest in the game is appreciated!

Here’s a link of a post describing the first game in the series.  

Check out the Character designs of the villagers/students here!

AC Fans, we’re hitting that one year mark of Animal Crossing New Leaf’s release. while the game never ends, by  all means, check this title out and see if its features catches your interest! 

If you would consider purchasing a 3DS E-Shop Downloadable Release of this game, please REBLOG this post to show These companies there is a strong interest and awareness of this game! Thank you for your time!

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Sonic Booms and the Doppler Effect

The Doppler effect is the shift in the frequency of a wave observed when the source of the wave (or the medium through which the wave travels) is moving relative to the observer.

We’re most familiar with the Doppler effect as it appears in sound waves traveling through air (i.e., pressure waves) – think of how the pitch of a siren drops as an emergency vehicle passes you. The first GIF shows a stationary source and the second shows a source moving to the right at 40% the speed of sound. Notice in the second GIF how the wavefronts are closer together in front of the source (producing a higher frequency) and further apart behind it (producing a lower frequency).

The Doppler effect is interesting in its own right, but things get much more exciting when the source travels at speeds greater than or equal to the speed of sound.

When the source travels at the speed of sound (GIF 3) the source will always be at the leading edge of the waves it produces, and when traveling faster than the speed of sound (GIF 4), the source will always be in front of the waves it produces. In both of these cases, notice how the waves overlap with each other. The high pressure areas of each wave constructively interfere and produce a region of extremely high pressure (much higher than in the surrounding areas). This rapid rise in air pressure is a shock wave, and the sound associated with it is a sonic boom.

In each of the GIFs above we see the radiating wavefronts on the left, and the pressure distribution and interference of the waves on the right.

Mathematica code posted here.

Additional source not linked above.

Simulating time travel: Doctor Who meets Professor Heisenberg
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(Phys.org)—University of Queensland researchers have simulated time travel using light particles. Lead author and PhD student Martin Ringbauer, from UQ’s School of Mathematics and Physics, said the study used photons – single particles of light – to simulate quantum particles traveling through time and study their behavior, possibly revealing bizarre aspects of modern physics. “The question of time travel features at the interface between two of our most successful yet incompatible physical theories – Einstein’s general relativity and quantum mechanics,” Mr Ringbauer said. “Einstein’s theory describes the world at the very large scale of stars and galaxies, while quantum mechanics is an excellent description of the world at the very small scale of atoms and molecules.” Einstein’s theory suggests the possibility of travelling backwards in time by following a space-time path that returns to the starting point in space, but at an earlier time-a closed timelike curve. This possibility has puzzled physicists and philosophers alike since it was discovered by Kurt Gödel in 1949, as it seems to cause paradoxes in the classical world, such as the grandparents paradox, where a time traveller could prevent their grandparents from meeting, thus preventing the time traveller’s birth. This would make it impossible for the time traveller to have set out in the first place. UQ Physics Professor Tim Ralph said it was predicted in 1991 that time travel in the quantum world could avoid such paradoxes. “The properties of quantum particles are ‘fuzzy’ or uncertain to start with, so this gives them enough wiggle room to avoid inconsistent time travel situations,” he said. Professor Ralph said there was no evidence that nature behaved in ways other than standard quantum mechanics predicted,but this had not been tested in regimes where extreme effects of general relativity played a role, such as near a black hole. (via Simulating time travel: Doctor Who meets Professor Heisenberg)

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