Results in Physics (Mar 2021)
On the semiclassical JWKB and MAF solutions of the uniformly lengthening pendulum via change of dependent variable in the Bessel’s equation
Abstract
Semiclassical JWKB method and method of modified Airy functions (MAF) are the fundamental conventional tools in quantum mechanical and optical waveguide systems and yet their classical mechanical applicability has somehow escaped much attention. In this work, semiclassical JWKB and MAF solution of a classical mechanical problem regarding the Lengthening Pendulum (LP) is being studied to reinforce the power of the semiclassical methods in this aspect. Here, JWKB and MAF solutions of the associated Bessel’s equation is being obtained by transforming it first into one of the normal forms via change of dependent variable on the contrary to our previous work where change of independent variable was used for the JWKB solution. By the same motivation for the suggested change of dependent variable here, we once again, obtain advantageous results in comparison to the conventional numerical exact solutions given in the literature where adiabatic LP systems with very low lengthening rates are required. We obtain both JWKB and MAF approximated Bessel functions successfully and our semiclassical solutions for the LP system works with a great accuracy even for much higher lengthening rates in comparison to these conventional numerical methods. Here we study under what conditions our semiclassical solutions can give accurate results by using the conventional JWKB and MAF applicability criteria. In effect, the semiclassical JWKB and MAF methods which is normally known to be used in quantum mechanical and optical waveguide systems are being used to solve the pure classical-mechanical LP system successfully and advantageously here and the applicability criteria of these semiclassical methods in terms of the physical LP system parameters are being determined and presented.
