Some Analytical Results in Dynamics of Spheroidal Galaxies

The surfaces of section in a harmonic oscillator potential, perturbed by quartic terms, are obtained analytically. A succession of action‐angle, Lissajous and Lie transformations near the 1:1 commensurability, reduces the three‐dimensional motion to a one‐dimensional one. The latter is solved in terms of Jacobi's elliptic functions. Existence conditions for periodic orbits are found and two general families of such solutions are introduced. Two examples of regular motions in oblate and prolate spheroids are discussed.