[Paper Review] Systematic approximations for the period of a finite amplitude pendulum

I. R. Gatland|ArXiv.org|Jul 6, 2007

Experimental and Theoretical Physics Studies7 references3 citations

TL;DR

This paper introduces a systematic method to improve the accuracy of period approximations for a finite amplitude pendulum by adjusting the final term of a truncated series expansion to account for omitted higher-order terms. The approach yields both lower and upper bounds on the true period, enabling tighter error estimates and more accurate intermediate approximations through systematic refinement of the series truncation.

ABSTRACT

The standard series expansion for the period of a finite amplitude pendulum as a function of energy (and hence amplitude) provides a lower limit on the period when the series is truncated. An adjustment to the last term in the truncated series to take account of the dropped terms improves the accuracy of the approximation and provides an upper limit on the period. More accurate approximations can then be obtained using intermediate expressions.

Motivation & Objective

• Address the inherent inaccuracy of truncated series expansions for the pendulum period at large amplitudes.
• Provide a systematic framework to refine period approximations beyond standard series truncation.
• Establish both lower and upper bounds for the true period using corrections to the truncated series.
• Enable more accurate intermediate approximations by leveraging the bounds as constraints.
• Improve practical utility of analytical approximations in classical mechanics and physics education.

Proposed method

• Utilizes the standard series expansion of the pendulum period in terms of elliptic integrals and amplitude.
• Truncates the series at a finite order, recognizing that this introduces a systematic underestimation of the true period.
• Applies a correction to the last retained term to account for the sum of all dropped higher-order terms.
• This correction yields an upper bound on the true period, complementing the truncated series' lower bound.
• Constructs intermediate approximations by combining the lower and upper bounds via weighted averages or rational functions.
• Employs numerical validation to confirm the improved accuracy and convergence behavior of the refined approximations.

Experimental results

Research questions

• RQ1 How can the accuracy of truncated series approximations for the pendulum period be systematically improved?
• RQ2 What correction strategy can be applied to the final term of a truncated series to account for omitted higher-order terms?
• RQ3 Can the corrected series provide both lower and upper bounds on the true pendulum period?
• RQ4 How do intermediate approximations derived from the bounds compare in accuracy to standard truncated series?
• RQ5 What is the quantitative improvement in error bounds and convergence rate using the proposed method?

Key findings

• The truncated series expansion for the pendulum period provides a lower bound on the true period due to the omission of positive higher-order terms.
• Adjusting the last term to include the sum of all dropped terms yields an upper bound on the true period.
• The resulting interval between the lower and upper bounds tightly brackets the true period, significantly reducing uncertainty.
• Intermediate approximations constructed from the bounds exhibit substantially improved accuracy compared to the standard truncated series.
• The method enables systematic refinement of the approximation without requiring higher-order series computation.
• Numerical validation confirms that the corrected approximations converge more rapidly and with smaller error than standard truncation.