Date of Award

Spring 5-29-2026

Document Type

Thesis (Undergraduate)

Department

Physics and Astronomy

First Advisor

Rufus Boyack

Abstract

The study of bounded and periodic motion in classical systems is of central importance in celestial mechanics, atomic models, and the theory of integrable systems. Central-force problems are of particular importance due to their rich geometric structure and the existence of conserved quantities that strongly constrain orbital behavior. In this thesis, we investigate the classical motion of bodies subject to two central-force potentials: the inverse-square and the pseudo-harmonic. For each system, explicit solutions for the radial and angular equations of motion are derived and analyzed. The resulting trajectories are classified according to whether they are bounded, unbounded, open, closed, or collapsing.

Motion of a body under an inverse-square potential separates into distinct regimes characterized either by outward escape trajectories with finite asymptotic angular displacement or by collapse into the origin at finite time. A radial harmonic term is added to the inverse-square potential to produce the pseudo-harmonic potential that we also analyze. We find that the pseudo-harmonic potential admits oscillatory radial behavior, periodic angular behavior, and, most interestingly, some closed and bounded orbits. These results are interpreted in the broader context of Bertrand’s theorem, which states that the only two potentials for which all bounded orbits are also closed are the attractive inverse-linear and radial harmonic potentials. The systems studied here thus provide useful examples of more general central-force dynamics lying outside the Bertrand class while still exhibiting rich and analytically tractable orbital structure.

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