Novel Approaches to Solve Simple Harmonic Motion

TL;DR

This paper introduces two innovative methods for solving simple harmonic motion, one based on force averaging and segmentation, and the other on recurrence relations, both converging to classical solutions.

Contribution

The paper presents two novel approaches to solve simple harmonic motion, extending traditional methods with segmentation and recurrence techniques.

Findings

Both methods accurately reproduce the period of simple harmonic motion.

The approaches converge to classical solutions in the large-N limit.

New techniques offer alternative perspectives for solving harmonic oscillations.

Abstract

This paper presents two novel approaches to solve the classic simple harmonic motion. In one approach, the distance between the equilibrium position and the maximal displacement is divided into N equal segments. In each segment, the mass moves with constant acceleration under the average of two forces at the ends of the segment. Summing up the time covering each segment and taking the large-N limit reproduce one quarter of the period for simple harmonic motion. In the other approach, the time moving from the maximal displacement to the equilibrium position is divided into N equal intervals. A recurrence relation for the displacement is obtained. The large-N limit of its solution results in the same solution as that obtained from solving differential equation.