A theoretical framework is developed for studying trajectories of ions reflected or leaked upstream from the earth's bow shock and subject solely to the Lorentz force in a steady interplanetary magnetic field B and the V¿B electric field. We include the effects of a sharp shock potential rise. Expressions are derived for the guiding center motion and gyromotion in a frame (the Hoffman-Teller frame) moving parallel to the shock surface with sufficient speed to transform the incident solar wind velocity into motion entirely along the interplanetary magnetic field: the appropriate equations are also provided to transform these motions back to the observer's frame. The utility of these expressions is illustrated by comparing the predicted upstream motions for four different source models for upstream ions: magnetic moment-conserving reflection of the solar wind ions, specular reflection of solar wind ions, magnetic moment-conserving leakage of magnetosheath ions, and leakage of magnetosheath ions parallel to the shock normal. This comparison reveals that, for identical geometries, the reflection models produce higher energies and/or gyromotion than do the leakage models. We further argue that in a single simple encounter with the shock, an ion should behave in an unmagnetized manner and hence should not conserve its magnetic moment. Conservation of magnetic moment, if it is to occur, would seem to require multiple encounters with the shock. We investigate the conditions under which such multiple encounters can occur and find that under most quasi-parallel geometries neither leaked nor reflected ions should probably conserve their magnetic moments.