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Non-Sinusoidal Harmonics and Special Functions
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
The history of Bessel functions did not just originate in the study of the wave and heat equations. These solutions originally came up in the study of the Kepler problem, describing planetary motion. According to G. N. Watson in his Treatise on Bessel Functions, the formulation and solution of Kepler’s Problem was discovered by Joseph-Louis Lagrange (1736–1813), in 1770. Namely, the problem was to express the radial coordinate and what is called the eccentric anomaly, E, as functions of time. Lagrange found expressions for the coefficients in the expansions of r and E in trigonometric functions of time. However, he only computed the first few coefficients. In 1816, Friedrich Wilhelm Bessel (1784–1846) had shown that the coefficients in the expansion for r could be given an integral representation. In 1824, he presented a thorough study of these functions, which are now called Bessel functions.
Numerical solution of ODE problems
Published in Alfio Borzì, Modelling with Ordinary Differential Equations, 2020
The Kepler problem is a classical problem of Hamiltonian dynamics with three invariants: the Hamiltonian (energy) function, the angular momentum, and the so-called Runge-Lenz vector. This problem is named after Johannes Kepler, known for his laws of planetary motion, and it usually refers to the motion of two point massive particles that interact through a gravitational force. In particular, in the case of bounded orbits, this motion consists of closed and periodic orbits.
Generalized Fourier Series and Function Spaces
Published in Russell L. Herman, An Introduction to Fourier Analysis, 2016
The history of Bessel functions did not just originate in the study of the wave and heat equations. These solutions originally came up in the study of the Kepler problem, describing planetary motion. According to G. N. Watson in his Treatise on Bessel Functions, the formulation and solution of Kepler's Problem was discovered by Joseph-Louis Lagrange (1736-1813), in
Numerical solution of Maxwell equations to study photothermal signals on a dielectric monolayer
Published in Journal of Modern Optics, 2019
Marco A. Molina-González, Arnulfo Castellanos-Moreno, Adalberto Corella-Madueño
A particle acted by a potential V(), described by classical mechanics, can have an analogy in geometrical optics by changing V() for the square of the refractive index. This formal relation has been used to study a problem in mechanics to translate the solution to an interesting system in optics (1–7). One of this is a spherical metallic particle inside a dielectric media, such that the particle is heated first by a laser. The refractive index is modified in each point (), such that the new system can be described by adding a new perturbative term to the dielectric function. This is proportional to 1/r, where r is the distance from the hot point to (). The analogy with classical mechanics is a particle whose motion occurs in a plane, the Kepler problem. Therefore, it is possible to think in a dielectric monolayer with a hot point that can be produced by some external mean, so that the photothermal detection can be studied.