Northrop and Teller have shown that when the guiding center motion of a magnetically trapped particle is averaged over its two fastest periodic modes, the result can be described by a Hamiltonian function, the Euler potentials (&agr;,&bgr;) of the particle's guiding field serving as canonical variables. In the present work the form of this Hamiltonian is developed for a dipole field, and the result is expressed by an analytic approximation accurate within about 1%. This allows extension of results derived for equatorial particles to particles with arbitrary pitch angles; in particular, it makes available even in the presence of electric fields orthogonal to the magnetic field a function K that is preserved by the bounce-averaged motion. This function provides at once the equations of drift paths in (&agr;,&bgr;), or their projections onto the equatorial plane; the derivation of a 'pacing function' u that times the progress of particles along such drift paths will also be described. Other applications include the derivation of accurate approximations of the mean drift velocity, the energization of electrically convected particles, and the 'bounce time' T.