September 14th, 2020

лошадь, диаграмма, Фейнман

Точный вывод скорости томасовской прецессии с помощью преобразований Лоренца

An exact derivation of the Thomas precession rate using the Lorentz transformation:

Masud Mansuripur
Using the standard formalism of Lorentz transformation of the special theory of relativity, we derive the exact expression of the Thomas precession rate for an electron in a classical circular orbit around the nucleus of a hydrogen-like atom.
Comments: 10 pages, 32 equations, 10 references
лошадь, диаграмма, Фейнман

Динамический хаос как модель чего-то там в бейсболе

Dynamical Chaos in a Simple Model of a Knuckleball:

Nicholas J. Nelson, Eric Strauss
The knuckleball is perhaps the most enigmatic pitch in baseball. Relying on the presence of raised seams on the surface of the ball to create asymmetric flow, a knuckleball's trajectory has proven very challenging to predict compared to other baseball pitches, such as fastballs or curveballs. Previous experimental tracking of large numbers of knuckleballs has shown that they can move in essentially any direction relative to what would be expected from a drag-only trajectory. This has led to speculation that knuckleballs exhibit chaotic motion. Here we develop a relatively simple model of a knuckleball that includes quadratic drag and lift from asymmetric flow which is taken from experimental measurements of slowly rotating baseballs. Our models can indeed exhibit dynamical chaos as long In contrast, models that omit torques on the ball in flight do not show chaotic behavior. Uncertainties in the phase space position of the knuckleball are shown to grow by factors as large as 106 over the flight of the ball from the pitcher to home plate. We quantify the impact of our model parameters on the chaos realized in our models, specifically showing that maximum Lyapunov exponent is roughly proportional to the square root of the effective lever arm of the torque, and also roughly proportional to the initial velocity of the pitch. We demonstrate the existence of bifurcations that can produce changes in the location of the ball when it reaches the plate of as much as 1.2 m for specific initial conditions similar to those used by professional knuckleball pitchers. As we introduce additional complexity in the form of more faithful representations of the empirical asymmetry force measurements, we find that a larger fraction of the possible initial conditions result in dynamical chaos.
Comments: 21 pages, 13 figures, accepted for publication in "Applied Mathematics and Computation"