The Harmonic Oscillator in the Plane and the Jordan-Schwinger Algebras
Abstract
The direct and indirect Lagrangian representations of the planar harmonic oscillator have been discussed. The reduction of these Lagrangians in their basic forms characterising either chiral, or pseudo - chiral oscillators have been given. A Hamiltonian analysis, showing its equivalence with the Lagrangian formalism has also been provided. Finally, we show that the chiral and pseudo - chiral modes act as dynamical structures behind the Jordan - Schwinger realizations of the SU(2) and SU(1,1) algebras. Also, the SU(1,1) construction found here is different from the standard Jordan - Schwinger form.
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.