Nonlinear effects associated with standing Alfv¿n waves may be significant in planetary magnetospheres. The field-aligned second-order force and acceleration generated by standing Alfv¿n waves in a dipolar magnetosphere are derived using previously published perturbation series for toroidal mode eigenfunctions and eigenvalues. The field-aligned ponderomotive force and ponderomotive acceleration (PMA) are then derived by time integration. The second-order acceleration and the PMA are independent of the plasma mass density distribution in the equatorial plane, and are only weakly dependent on the density structure along magnetic field lines. The second-order acceleration has an oscillation at twice the standing wave frequency superimposed on the PMA at a given position. The amplitude and phase of this oscillation are dependent on ionospheric conductance and field-aligned density structure. The PMA has an approximate L5 dependence, and can have relatively large values in the outer magnetosphere. For example a fundamental standing Alfv¿n wave with equatorial electric field amplitude 5 mV m-1 gives a maximum PMA of 12g toward the equatorial plane at L=9. An approximate upper limit on the energy of particles accelerated by this mechanism appears to be of the order of 1 keV, attained by O+ ions accelerated from the ionosphere to the equatorial plane at high L values.