Chaos theory A double rod pendulum animation showing chaotic behavior. Starting the pendulum from a slightly different initial condition would result in a completely different trajectory. The double rod pendulum is one of the simplest dynamical systems that has chaotic solutions. Chaos: When the present determines the future, but the approximate present does not approximately determine the future. Chaotic behavior can be observed in many natural systems, such as weather and climate.[6][7] This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Introduction[edit] Chaos theory concerns deterministic systems whose behavior can in principle be predicted. Chaotic dynamics[edit] The map defined by x → 4 x (1 – x) and y → x + y mod 1 displays sensitivity to initial conditions. In common usage, "chaos" means "a state of disorder".[9] However, in chaos theory, the term is defined more precisely. where , and , is: .
Popper's experiment Popper's experiment is an experiment proposed by the philosopher Karl Popper. As early as 1934 he was suspicious of, and was proposing experiments to test, the Copenhagen interpretation, a popular subjectivist interpretation of quantum mechanics.[1][2] Popper's experiment is a realization of an argument similar in spirit to the thought experiment of Einstein, Podolsky and Rosen (the EPR paradox) although not as well known. There are various interpretations of quantum mechanics that do not agree with each other. Despite their differences, they are experimentally nearly indistinguishable from each other. The most widely known interpretation of quantum mechanics is the Copenhagen interpretation put forward by Niels Bohr. It says that observations lead to a wavefunction collapse, thereby suggesting the counter-intuitive result that two well separated, non-interacting systems require action-at-a-distance. Popper's proposed experiment[edit] Popper wrote: The debate[edit] They concluded that: . .
Falling cat problem A falling cat modeled as two independently rotating parts turns around while maintaining zero net angular momentum The falling cat problem consists of explaining the underlying physics behind the common observation of the cat righting reflex: how a free-falling cat can turn itself right-side-up as it falls, no matter which way up it was initially, without violating the law of conservation of angular momentum. Although somewhat amusing, and trivial to pose, the solution of the problem is not as straightforward as its statement would suggest. The apparent contradiction with the law of conservation of angular momentum is resolved because the cat is not a rigid body, but instead is permitted to change its shape during the fall. The behavior of the cat is thus typical of the mechanics of deformable bodies. See also[edit] References[edit] Arabyan, A; Tsai, D. (1998), "A distributed control model for the air-righting reflex of a cat", Biol. Further reading[edit]
Dirac Theory Next: Smaller EffectsUp: Everything You Always Wanted Previous: Kinetic Energy Correction The theory of Paul Dirac represents an attempt to unify the theories of quantum mechanics and special relativity. That is, one seeks a formulation of quantum mechanics which is Lorentz invariant, and hence consistent with special relativity. For a free particle, relativity states that the energy is given by . If H and p are associated with the same operators as in Schrödinger theory, then one expects the wave equation This is known as the Klein-Gordan Equation. However, this creates a new problem. With this form of the Hamiltonian, the wave equation can be written In order for this to be valid, one hopes that when it is squared the Klein-Gordan equation is recovered. and are at least matrices and the wavefunction is a four-component column matrix. It turns out that equation 63 describes only a particle with spin 1/2. Including the potential now in the Hamiltonian, equation 63 becomes where . can be written
100 Time-Saving Search Engines for Serious Scholars While burying yourself in the stacks at the library is one way to get some serious research done, with today’s technology you can do quite a bit of useful searching before you ever set foot inside a library. Undergraduates and grad students alike will appreciate the usefulness of these search engines that allow them to find books, journal articles and even primary source material for whatever kind of research they’re working on and that return only serious, academic results so time isn’t wasted on unprofessional resources. Note: Visit our updated list for the latest in academic search engines. General Start off your research with one of these more general academic search engines. Intute: Use this website’s search tools to find the best and most reliable sites to start your research. Meta Search Want to search it all at once? Dogpile: Search Google, Yahoo, Bing and more at once with this great search engine. Databases and Archives Books and Journals Science Math and Technology Social Science
Bohr–Einstein debates The Bohr–Einstein debates were a series of public disputes about quantum mechanics between Albert Einstein and Niels Bohr, who were two of its founders. Their debates are remembered because of their importance to the philosophy of science. An account of the debates has been written by Bohr in an article titled "Discussions with Einstein on Epistemological Problems in Atomic Physics".[1] Despite their differences of opinion regarding quantum mechanics, Bohr and Einstein had a mutual admiration that was to last the rest of their lives.[2] Pre-revolutionary debates[edit] Einstein was the first physicist to say that Planck's discovery of the quantum (h) would require a rewriting of physics. 1913 brought the Bohr model of the hydrogen atom, which made use of the quantum to explain the atomic spectrum. The quantum revolution[edit] Einstein rejected this interpretation. Post-revolution: First stage[edit] Figure A. Einstein's slit. Figure C. Figure D. . which satisfies the relation: . .
Bell test experiments Bell test experiments or Bell's inequality experiments are designed to demonstrate the real world existence of certain theoretical consequences of the phenomenon of entanglement in quantum mechanics which could not possibly occur according to a classical picture of the world, characterised by the notion of local realism. Under local realism, correlations between outcomes of different measurements performed on separated physical systems have to satisfy certain constraints, called Bell inequalities. John Bell derived the first inequality of this kind in his paper "On the Einstein-Podolsky-Rosen Paradox".[1] Bell's Theorem states that the predictions of quantum mechanics cannot be reproduced by any local hidden variable theory. The term "Bell inequality" can mean any one of a number of inequalities satisfied by local hidden variables theories; in practice, in present day experiments, most often the CHSH; earlier the CH74 inequality. Conduct of optical Bell test experiments[edit] Pan et al.'
Cat physics – and we are not making this up Cats may skulk, and cats may fall – but no matter what they do, cats must obey the laws of physics. Scientists have tried repeatedly to figure out how they manage to do it. At the extreme, physicists analysed what happens to a dropped cat. That's a cat in free-fall, a cat hurtling earthwards with nothing but kitty cunning to keep it from crashing. In 1969, TR Kane and MP Scher of Stanford University, in California, published a monograph called A Dynamical Explanation of the Falling Cat Phenomenon. It remains one of the few studies about cats ever published in the International Journal of Solids and Structures. "It is well known that falling cats usually land on their feet and, moreover, that they can manage to do so even if released from complete rest while upside-down … numerous attempts have been made to discover a relatively simple mechanical system whose motion, when proceeding in accordance with the laws of dynamics, possesses the salient features of the motion of the falling cat.
Fermat's Last Theorem The 1670 edition of Diophantus' Arithmetica includes Fermat's commentary, particularly his "Last Theorem" (Observatio Domini Petri de Fermat). In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. Overview[edit] The claim eventually became one of the most notable unsolved problems of mathematics. The Pythagorean equation has an infinite number of positive integer solutions for a, b, and c; these solutions are known as Pythagorean triples. Subsequent developments and solution[edit] Mathematical history[edit] Pythagoras and Diophantus[edit] Pythagorean triples[edit] Examples of Pythagorean triples include (3, 4, 5) and (5, 12, 13).
Khan Academy General relativity General relativity, or the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1916[1] and the current description of gravitation in modern physics. General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations. Some predictions of general relativity differ significantly from those of classical physics, especially concerning the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light. Einstein's theory has important astrophysical implications. History[edit] Albert Einstein developed the theories of special and general relativity.
Elitzur–Vaidman bomb tester Bomb-testing problem diagram. A - photon emitter, B - bomb to be tested, C,D - photon detectors. Mirrors in the lower left and upper right corners are half-silvered. In physics, the Elitzur–Vaidman bomb-testing problem is a thought experiment in quantum mechanics, first proposed by Avshalom Elitzur and Lev Vaidman in 1993.[1] An actual experiment demonstrating the solution was constructed and successfully tested by Anton Zeilinger, Paul Kwiat, Harald Weinfurter, and Thomas Herzog from the University of Innsbruck, Austria and Mark A. Problem[edit] Consider a collection of bombs, of which some are duds. Solution[edit] Start with a Mach–Zehnder interferometer and a light source which emits single photons. When a photon's state is non-deterministically altered, such as interacting with a half-silvered mirror where it non-deterministically passes through or is reflected, the photon undergoes quantum superposition, whereby it takes on all possible states and can interact with itself. P.
If my Pussy smells like Tuna, why doesn't my Cat eat me out? It's a legitimate question, don't you think? I recently tickled my kitty in front of my cat to answer this question; a question every girl who has ever been mistaken for a fish and chip shop by a blind man has pondered, but has been afraid to ask. Initially I felt guilty about my little experiment, but curiosity won hands down—or possibly fingers in—and I was quickly so horny I didn't really give a bugger. At first my cat was none the wiser as I slyly slid out of my slightly damp knickers beneath the sheets and reached up for my vibrator. At first I was reluctant to turn it on as Lulu was literally six inches from my waist, so why draw attention to what I was doing if I didn't have to, right? Well I had to—this was a scientific experiment, remember? Meanwhile my inquisitive moggie was beginning to suspect something sinister was going on—or possibly round and round— beneath the twitching sheets. She probably wondered what all that grunting was about.