Symmetry reduction for central force problems

TL;DR

This paper illustrates symmetry reduction in central force problems using phase portraits and compares it to symplectic reduction in free rigid body dynamics, highlighting the geometric structure of the reduced systems.

Contribution

It provides an elementary illustration of symmetry reduction for central force problems and visualizes the reduced dynamics through phase portraits.

Findings

Phase portraits of reduced dynamics are obtained as intersections of Casimir and energy level sets.

Symmetry reduction techniques are applied to central force problems, illustrating their geometric structure.

Comparison made between symmetry reduction in central force problems and free rigid body dynamics.

Abstract

We given an elementary illustration of symmetry reduction for central force problems, drawing phase portraits of the reduced dynamics as the intersection of Casimir and energy level sets in three dimensions. These systems form a classic example of symplectic reduction which can usefully be compared to the more-commonly seen case of the free rigid body.