  We consider the dynamics of a Hamiltonian particle forced by a rapidly
oscillating potential in $\dim$-dimensional space. As alternative to the
established approach of averaging Hamiltonian dynamics by reformulating the
system as Hamilton-Jacobi equation, we propose an averaging technique via
reformulation using the Maupertuis principle. We analyse the result of these
two approaches for one space dimension. For the initial value problem the
solutions converge uniformly when the total energy is fixed. If the initial
velocity is fixed independently of the microscopic scale, then the limit
solution depends on the choice of subsequence. We show similar results hold for
the one-dimensional boundary value problem. In the higher dimensional case we
show a novel connection between the Hamilton-Jacobi and Maupertuis approaches,
namely that the sets of minimisers and saddle points coincide for these
functionals.
