One such example is the problem of cyclotron motion -- the motion of a charged particle in a constant magnetic field. A readily solved problem in both its relativistic and nonrelativistic forms, it made its appearance in the earliest literature of wave mechanics: in papers by Darwin (59), and Kennard (60) in 1927. Later on, around 1930, there was a period of rather intense interest in the quantum mechanical behavior of magnetically deflected electrons which can be attributed to the fact that in those days there were well defined discrepancies in the experimental values of 
, the charge-to-mass ratio of the electron, and even for 
itself, as deduced from diverse experiments.
The classical treatment of the cyclotron problem (which was hardly yet known by that name) yielded a circular orbit whose radius was inversely proportional to the magnetic field strength, but Page (61), in 1930 had considered the possibility that the radius of curvature of the trajectory of the quantum mechanical wave packet might be different from the classical radius, and that the discrepancy might influence the value of in the right direction. Unfortunately his solutions tended to bear out this point of view, but were quickly challenged by various authors, and a certain exchange of opinions followed. Uhlenbeck and Young (62) disputed the result; Landau (63) had considered the problem in his theory of diamagnetism in metals; Alexandrow (64) had given a solution; and it was also considered from the relativistic point of view by Plesset (65) and by Huff (66), who in 1931 finally pinpointed Page's error as having overlooked some of the possible solutions whose absence from the wave packet would bias its average radius of curvature. Whereas a definite solution was obtained for this nonrelativistic problem, the actual solution of the radial part of Dirac's relativistic equation for cyclotron motion seems to have been gotten by approximate methods; In 1932, Laporte (67) discussed an application of the WKB method to obtain such a solution. Even in the present day there remains some discussion (68). Rabi (69) gave one of the first solutions for the Dirac equation with a uniform magnetic field.
After a lapse of some 20 years, the problem was again revived in a series of papers by Johnson and Lippman using the elegant operator techniques much exploited by Schwinger. They considered both nonrelativistic motion (70), in 1949, and relativistic motion (71), in 1950. They were explicitly concerned with constants of the motion, although they still did not mention the possible symmetries. Among their results were a pair of constants of the motion which located the center of the cyclotron orbit, but which obeyed the commutation rules of a pair of conjugate ladder operators. Thus, both coordinates of the center could not be simultaneously observed. The energy levels were infinitely degenerate, corresponding to the freedom to locate the center of the orbit anywhere in the plane perpendicular to the magnetic field.
The Hamiltonian for the cyclotron problem is
One thereby obtains a bilinear function of the canonical coordinates and momenta, which can be simplified considerably by transforming to a rotating coordinate system. This is the treatment generally used in engineering discussions of the magnetron, and allied literature. Unfortunately the latter articles rarely use even the vector potential, and are in no way concerned with symmetry or degeneracy problems. Nevertheless, the constants of the motion which are found in the rotating coordinate system carry over to the stationary system, and it is quite advantageous to be able to use the Hamiltonian of the rotating system. In any event, when one deals with a bilinear Hamiltonian, the Hamiltonian equations of motion become linear equations with constant coefficients, to which special methods may be applied. These aspects were discussed by Dulock and McIntosh (72) in 1966 when they superimposed a harmonic oscillator potential in order to remove the degeneracy arising from the arbitrariness of the positioning of the origin, and emphasize the resulting symmetry. Pure cyclotron motion as well as pure harmonic oscillator motion were obtained as limiting cases.
The possibility of perturbing the harmonic oscillator with a uniform magnetic field was noticed as long ago as 1928 (73). The combination is mathematically equivalent to an anisotropic harmonic oscillator, for which one expects to obtain bounded closed orbits only for certain combinations of magnetic field strength and harmonic oscillator force constant. Those would be the combinations which result in rational frequency ratios of the oscillator. In the limit of pure cyclotron motion one finds that the unitary unimodular Lie group appropriate to the harmonic oscillator contracts to von Neumann's algebra of ladder operators; and in addition to the quadratic constants of the motion typical of the harmonic oscillator one finds that there are even constants of the motion which are linear in the coordinates and momenta.
Although the problem of cyclotron motion is really a rather simple one, particularly the nonrelativistic version serves to illustrate principles which are considerably obscured in more complex systems. First of all, a uniform magnetic field has an apparent translational symmetry which is not shared by its vector potential, in whatever gauge. This means that the Hamiltonian is not translationally invariant, even though it yields equations of motion with such an invariance. Of course, the change of gauge arising from translation must be incorporated in the constants of the motion generating infinitesimal translations, with the result that Johnson and Lippman's constants are not simply the momenta, but depend on the coordinates as well. This same phenomonon appears in crystal lattices which are subjected to a magnetic field, and requires the use of magnetic space groups to properly accommodate the phase change in the wave function arising from the gauge transformation. As one sees in dealing with the symmetry of a magnetic monopole, rotational symmetries in the presence of a magnetic field are also modified by an infinitesimal gauge transformation, but such problems can be avoided with the cylindrically symmetric uniform magnetic field of the cyclotron motion.
An additional possible source of complications fortunately does not arise in the cyclotron problem; it is convenient to transform the equations to a rotating coordinate system to simplify the Hamiltonian equations and obtain their solutions. Had constants of the motion somehow arisen, which were explicitly time dependent in the original coordinates, one would only have obtained constant rates and not constants of the motion in the original coordinate system. Relativistic cyclotron motion has been particularly useful for the insight which it affords to the symmetries of the Dirac equation.