On the MAF solution of the uniformly lengthening pendulum via change of independent variable in the Bessel’s equation

Abstract

Abstract Common recipe for the Lengthening Pendulum (LP) involves some change of variables to give a relationship with the Bessel’s equation. In this work, semiclassical MAF (Modified Airy Function) solution of the LP is being obtained by first transforming the related Bessel’s equation into the normal form via the suggested change of independent variable just as one of our recent work regarding the JWKB solution of the LP in (Deniz, 2017). MAF approximation of the first order Bessel Functions (ν = 1) of both type along with their zeros are being obtained analytically with a very good accuracy as a result of the appropriately chosen associated initial values and they are extended to the neighbouring orders (ν = 0 and 2) by the recursion relations. Although common numerical methods given in the literature require adiabatic LP systems where the lengthening rate is small, MAF solution presented here can safely be used for higher lengthening rates and a criterion for its validity is determined via the use of MAF applicability criterion given in the literature. As a result, the semiclassical MAF method which is normally used for the quantum mechanical and optical waveguide systems is applied to the classical LP system successfully just as our previous work regarding the JWKB solution of the LP. Interestingly, we have very accurate results in the entire domain except for x ≈ 0 .