The symmetry of the force field of a magnetic monopole is comparable in its simplicity to that of the hydrogen atom or a harmonic oscillator. Both these latter systems possess a ``hidden'' symmetry which leads to an ``accidental'' degeneracy in the energy spectrum of their Schrödinger equations. Since the monopole field is derived from a vector potential which is not symmetric, but undergoes a gauge transformation under rotation, the concepts of symmetry and constants of the motion must be expressed properly in the presence of velocity-dependent forces. It is found that neither the mechanical nor the canonical angular momentum is conserved in the presence of a monopole field, but rather a total angular momentum which incorporates angular momentum resident in the magnetic field. The total angular momentum defines a cone, on whose surface the motion takes place, whatever central electrostatic potential may exist. Neither the harmonic oscillator nor the hydrogen atom retain their accidental degeneracy when the nucleus possesses a magnetic charge, but if a repulsive centrifugal potential proportional to the square of the pole strength is added, accidentally degenerate systems with a higher symmetry result. The symmetry of such a harmonic oscillator is still somewhat obscure, but the ``charged Coulombic monopole'' has an O(4) symmetry group generated by the total angular momentum together with a Runge vector constructed from the total angular momentum. The irreducible representations of O(4) which occur are not the n2 representations of the hydrogen atom, but the m · n (m − n = 2ε, twice the monopole charge) representations which cannot be realized by four-dimensional spherical harmonics. The magnetic pole strength must be quantized, if admissible solutions of Schrödinger's equation are to exist, and according to Schwinger's quantization (ε = nℏc/e) if the wavefunctions are to be single valued; in any event, the ground state will be degenerate.

A paradox has arisen from some recent treatments of accidental degeneracy which claim that, for three degrees of freedom, SU(3) should be a universal symmetry group. Such conclusions are in disagreement both with experimentally observed spectra and with the generally accepted solutions of Schrödinger's equation. The discrepancy occurs in the transition between classical and quantum-mechanical formulations of the problems, and illustrates the care necessary in forming quantum-mechanical operators from classical expressions. The hydrogen atom in parabolic coordinates in two dimensions, for which the traditional treatment of Fock, extended by Alliluev, requires the symmetry group O(3), is a case for which the newer methods of Fradkin, Mukunda, Dulock, and others require SU(2). Although these groups are only slightly different, SU(2) fails to be the ``universal'' symmetry group on account of the multiple-valuedness of the parabolic representation. This conclusion extends a result of Han and Stehle: that, for rather similar reasons, SU(2) cannot be the classical symmetry group for the two-dimensional hydrogen atom.

Rule 54, a two state, three neighbor cellular automaton in Wolfram's systems of nomenclature, is less complex that Rule 110, but nevertheless possess a rich and complex dynamics. We provide a systematic and exhaustive analysis of glider behavior and interactions, including a catalog of collisions. Many o f t h e m s h o ws promise are computational elements.

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A further study of the degeneracy of the two dimensional harmonic oscillator is made, both in the isotropic and anisotropic cases. By regarding the Hamiltonian as a linear operator acting through the Poisson bracket on functions of the coordinates and momenta, a method applicable generally to bilinear Hamiltonians, it is shown how all possible rational constants of the motion may be generated. They are formed from eigenfunctions of the Hamiltonian which are linear combinations of the coordinates and momenta, and which belong to negative pairs of eigenvalues. Canonical coordinates, which may be visualized geometrically for the isotropic oscillator in terms of the Hopf mapping, place the symmetry group responsible for the accidental degeneracy clearly in evidence. Surprisingly, one finds that the unitary unimodular group SU2, is the symmetry group in all cases, even including that of an anisotropic oscillator with incommensurable frequencies. The lack of a quantum-mechanical analogy in the latter case is due to a lack of the necessary transcendental roots of the operators involved in attempting to use the correspondence principle, rather than to the lack of a symmetry group for the classical problem.

The theory of accidental degeneracy is surveyed, particular attention being paid to the connection between the accidental degeneracy of the two-dimensional isotropic harmonic oscillator and the theory of angular momentum.

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