Finite oscillator models obey the same dynamics as the classical and quantum oscillators, but the operators corresponding to position, momentum, Hamiltonian, and angular momentum are generators of the compact Lie group SO(D), and form the Lie algebra so(D). One-dimensional finite oscillators, shallow planar optical waveguides, and finite data sets are so(3) systems; and two-dimensional finite oscillators, shallow cylindrical waveguides, and pixellated screens are governed by so(4). The physical reinterpretation of the generators of these algebras as observables that take a finite number of values, fits in a coherent picture of their phase space.

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