Abstract
In this paper, a new analytical solution to the undamped Helmholtz oscillator equation in terms of the Weierstrass elliptic function is reported. The solution is given for any arbitrary initial conditions. A comparison between our new solution and the numerical approximate solution using the Range Kutta approach is performed. We think that the methodology employed here may be useful in the study of several nonlinear problems described by a differential equation of the form z ″ = F z in the sense that z = z t . In this context, our solutions are applied to some physical applications such as the signal that can propagate in the LC series circuits. Also, these solutions were used to describe and investigate some oscillations in plasma physics such as oscillations in electronegative plasma with Maxwellian electrons and negative ions.
Highlights
Descriptions of many of them date back to, at least, 1892, when the book by Greenhill [1] appeared, presenting a variety of such problems: a simple pendulum, catenaries, and a uniform chain that rotates steadily with a constant angular velocity about an axis to which the chain is fixed at two points
As a contribution to the literature, in this article, we present the exact solution to the Helmholtz Oscillator for the given arbitrary initial conditions by means of the Weierstrass elliptic function
Equation (1) is applied for mathematical modeling in physics and engineering like general relativity, betatron oscillations, vibrations of shells, vibrations of the acoustically driven human eardrum, and solid-state physics [24,25,26,27,28]. e Helmholtz oscillator can be interpreted as a particle moving in a quadratic potential field, and it has been studied in a nonlinear circuit theory
Summary
Descriptions of many of them date back to, at least, 1892, when the book by Greenhill [1] appeared, presenting a variety of such problems: a simple pendulum, catenaries, and a uniform chain that rotates steadily with a constant angular velocity about an axis to which the chain is fixed at two points. As a contribution to the literature, in this article, we present the exact solution to the Helmholtz Oscillator for the given arbitrary initial conditions by means of the Weierstrass elliptic function. We will derive the exact solution to the Helmholtz oscillator: Mathematical Problems in Engineering q€ + α + βq + cq 0, q(0) q0,. Equation (1) is applied for mathematical modeling in physics and engineering like general relativity, betatron oscillations, vibrations of shells, vibrations of the acoustically driven human eardrum, and solid-state physics [24,25,26,27,28]. E Helmholtz oscillator can be interpreted as a particle moving in a quadratic potential field, and it has been studied in a nonlinear circuit theory. One of the possible interesting interpretations of equation (1) is given by a simple electrical circuit
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