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Exploring the action landscape with trial world-lines

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 the action landscape in the space of nearby world-lines. Although these trial world-lines form only a subset of the space of all nearby world-lines, they provide a pedagogical tool that complements the traditional Lagrange equation approach and is well-suited for advanced undergraduate students.