In order to analyse in full detail the dynamics of a charged particle in the field of a magnetic
dipole, we propose to study the restricted motion of the particle in a spherical surface with the
dipole at its centre. This model can be considered as the classical non-relativistic St?rmer problem
within a sphere, and although this problem no longer represents the real St?rmer problem, it shows
the complex behaviour of this magnetic field through the classical dynamics equations that can be
formally integrated. We start from a Lagrangian approach which allows us to analyse the dynamical
properties of the system, such as the role of a velocity dependent potential, the symmetries and the
conservation properties. We derive the Hamilton equations of motion, which in this restricted case
can be reduced to a quadrature. From the Hamiltonian function we find, for the polar angle, an
equivalent one-dimensional system of a particle in the presence of an effective potential. This
equivalent p…