  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.
