Exploring the action landscape with trial world-lines
Yogesh N. Joglekar, Weng Kian Tham

TL;DR
This paper visualizes the action landscape around classical world-lines for basic physics problems, offering a pedagogical tool that enhances understanding of the Hamilton action principle for advanced undergraduates.
Contribution
It introduces explicit trial world-lines with analytical action expressions, enabling graphical visualization of the action landscape in introductory physics problems.
Findings
Visualized the action landscape for free, accelerating, and harmonic oscillator cases.
Provided analytical expressions for trial world-lines and their actions.
Enhanced pedagogical understanding of the Hamilton principle.
Abstract
The Hamilton action principle, also known as the principle of least action, and Lagrange equations are an integral part of advanced undergraduate mechanics. At present, substantial efforts are ongoing to suitably incorporate the action principle in introductory physics courses. Although the Hamilton principle is oft stated as "the action for any nearby trial world-line is greater than the action for the classical world-line", the landscape of action in the space of world-lines is rarely explored. Here, for three common problems in introductory physics - a free particle, a uniformly accelerating particle, and a simple harmonic oscillator - we present families of trial world-lines, characterized by a few parameters, that evolve continuously from their respective classical world-lines. With explicit analytical expressions available for the action, they permit a graphical visualization of…
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