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

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, conventional semiclassical JWKB solution (named after Jeffreys, Wentzel, Kramers and Brillouin) 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’. JWKB approximation of the first-order Bessel functions (ν=1) of both types 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. The required initial values are also being studied and a quantization rule regarding the experimental LP parameters is being determined. Although common numerical methods given in the literature require adiabatic LP systems where the lengthening rate is slow, JWKB solution presented here can safely be used for higher lengthening rates and a criterion for its validity is determined by the JWKB applicability criterion given in the literature. As a result, the semiclassical JWKB method which is normally used for the quantum mechanical and optical waveguide systems is applied to the classical LP system successfully.

Appendix A

To illustrate, as an alternative two-experiment set (one for finding J ₁ and the other for finding Y ₁) where only l ₀ is allowed to change among the set {g,l ₀,v}, let us consider the same system in (6.2). But now we have: {g = g ^′ ∧$l_{0}^{\prime }$ ≠ l ₀ ∧ v ^′ = v} [34d], and let us consider different initial values as different alternatives, say: $x_{0}^{\prime }\neq x_{0}\Rightarrow \{ \alpha _{1}^{\prime }\neq \alpha _{1}\neq 0, \beta _{1}^{\prime }$ = β ₁ = 0}. According to Lemma 4.2, we do not have to choose the experimental initial values necessarily as the nth zeros of Y ₁ and J ₁ as we did in (4.17f)–(4.18f) to obtain the desired Bessel functions: J ₁ ≈$\tilde {J}_{1}$ and Y ₁ ≈$\tilde {Y}_{1}$ entirely. Here, although we now choose different initial values in our alternative two-experiment set, in effect, we should have the same JWKB-approximated Bessel functions if Lemma 4.2 without requiring the nth root of Y ₁ and J ₁ is true. Choosing such an x ₀ point on J ₁ in part (i) and such an $x_{0}^{\prime }$ point on Y ₁ in part (ii) according to Lemma 4.2 with (4.16a) and (4.16b), we have the following experimental details giving the same results as we obtained above:

(i) For finding $\tilde {J}_{1}(x)~({\approx } J_{1}(x))$ via Lemma 4.2:

(ii) For finding $\tilde {Y}_{1}(x)~({\approx } Y_{1}(x))$ via Lemma 4.2: