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We study the drift of slow variables in a slow-fast Hamiltonian system withseveral fast and slow degrees of freedom. For any periodic trajectory of thefast subsystem with the frozen slow variables we define an action. For a familyof periodic orbits, the action is a scalar function of the slow variables andcan be considered as a Hamiltonian function which generates some slow dynamics.These dynamics depend on the family of periodic orbits.  Assuming the fast system with the frozen slow variables has a pair ofhyperbolic periodic orbits connected by two transversal heteroclinictrajectories, we prove that for any path composed of a finite sequence of slowtrajectories generated by action Hamiltonians, there is a trajectory of thefull system whose slow component shadows the path.

Drift of slow variables in slow-fast Hamiltonian systems