Search Results (2)

In this work, we investigate the transition from regular dynamics to chaotic behavior in a one-dimensional quartic anharmonic classical oscillator driven by a time-dependent external square-wave force. Owing to energy conservation, the motion of an undriven quartic anharmonic oscillator is regular, periodic, and stable. For a driven quartic anharmonic oscillator, the equations of motion cannot be solved analytically due to the presence of an anharmonic term in the potential energy function. Using the fourth-order Runge–Kutta method to numerically solve the equations of motion for the driven quartic anharmonic oscillator, we find that the oscillator motion under the influence of a sufficiently small driving force remains regular, while by gradually increasing the driving force, a series of nonlinear resonances can occur, grow, overlap, and ultimately disappear due to the emergence of chaos. Full article

Among the few exactly solvable problems in theoretical physics, the 2D (two-dimensional) Newtonian free fall problem in Euclidean space is perhaps the least known as compared to the harmonic oscillator or the Kepler–Coulomb problems. The aim of this article is to revisit this problem at the classical level as well as the quantum level, with a focus on its dynamical symmetries. We show how these dynamical symmetries arise as a special limit of the dynamical symmetries of the Kepler–Coulomb problem, and how a connection to the quartic anharmonic oscillator problem, a long-standing unsolved problem in quantum mechanics, can be established. To this end, we construct the Hilbert space of states with free boundary conditions as a space of square integrable functions that have a special functional integral representation. In this functional space, the free fall dynamical symmetry algebra is shown to be isomorphic to the so-called Klink’s algebra of the quantum quartic anharmonic oscillator problem. Furthermore, this connection entails a remarkable integral identity for the quantum quartic anharmonic oscillator eigenfunctions, which implies that these eigenfunctions are in fact zonal functions of an underlying symmetry group representation. Thus, an appropriate representation theory for the 2D Newtonian free fall quantum symmetry group may potentially open the way to exactly solving the difficult quantization problem of the quartic anharmonic oscillator. Finally, the initial value problem of the acoustic Klein–Gordon equation for wave propagation in a sound duct with a varying circular section is solved as an illustration of the techniques developed here. Full article