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We consider the dynamics of a Hamiltonian particle forced by a rapidlyoscillating potential in $\dim$-dimensional space. As alternative to theestablished approach of averaging Hamiltonian dynamics by reformulating thesystem as Hamilton-Jacobi equation, we propose an averaging technique viareformulation using the Maupertuis principle. We analyse the result of thesetwo approaches for one space dimension. For the initial value problem thesolutions converge uniformly when the total energy is fixed. If the initialvelocity is fixed independently of the microscopic scale, then the limitsolution depends on the choice of subsequence. We show similar results hold forthe one-dimensional boundary value problem. In the higher dimensional case weshow a novel connection between the Hamilton-Jacobi and Maupertuis approaches,namely that the sets of minimisers and saddle points coincide for thesefunctionals.

Effective Hamiltonian Dynamics via the Maupertuis Principle