### Abstract

Many Hamiltonian systems possess special families of solutions which can be described approximately as slowly drifting periodic orbits. Examples include the gravitational three-body problem, the interaction of two identical charged particles in a magnetic field, and the propagation and interaction of discrete breathers (time-periodic spatially localised motions in Hamiltonian networks of units). Theory will be presented for how to improve a zeroth order manifold of approximate solutions to r-th order for any r > 0, meaning one that contains all nearby periodic orbits of nearby period and has error of order the r-th power of the drift field (even with a small constant for r=1). In the normally hyperbolic case an exactly invariant nearby submanifold can be constructed. If there are normally elliptic directions, however, this is impossible in general but the above r-th order approximations can be achieved provided that the normal frequencies avoid all multiples of that of the approximately periodic motion and are fast compared with the drift. An effective Hamitlonian is derived to describe the drift of the orbits. Applications to the above fields will be given. An introduction has been published as R.S.MacKay, Slow Manifolds, in Energy localisation and transfer, eds T.Dauxois, A.Litvak-Hinenzon, R.S.MacKay, A.Spanoudaki (World Sci Publ Co, 2004), 149-192.