Motivated by the possibility of collective acceleration of ions trapped in an accelerating space‐charge wave on a strongly magnetized electron beam, the ion trajectories in such a configuration are studied. The motions perpendicular and parallel to the beam direction are coupled by a nonlinear term in the ion Hamiltonian that is proportional to the wave amplitude. Because of this coupling, the motion deviates markedly from that of a linear harmonic oscillator in certain resonant regions of phase space. A sequence of canonical transformations is used to study the motion in these regions. It is shown that wave amplitudes that are too small to trap beam electrons are too small to cause these reasonances to overlap. In the absence of such overlap, the motion is not discernibly ergodic in any three‐dimensional subspace of the energy hypersurface bcause there exists a third constant of the motion in addition to the total energy and angular momentum. These conclusions are verified using surface‐of‐section techniques to study numerically integrated ion trajectories. It is observed that the third constant of the motion constrains on ion initially trapped in a potential well of the wave to remain trapped in the well. Therefore, within the bounds of the physical model presented here, ergodic behavior poses no threat to attempts at collective ion acceleration in space‐charge waves on a electron beam.