  We present a coordinate-free approach for constructing approximate first
integrals of generalized slow-fast Hamiltonian systems, based on the global
averaging method on parameter-dependent phase spaces with $\mathbb{S}^1
-$symmetry. Explicit global formulas for approximate second-order first
integrals are derived. As examples, we analyze the case quadratic in the fast
variables (in particular, the elastic pendulum), and the charged particle in a
slowly-varying magnetic field.
