There is developed an approach to studying ergodic properties of time-dependent periodic Hamiltonian flows on symplectic metric manifolds having applications in mechanics and mathematical physics. Based both on J. Mather’s [9] results about homology of probability invariant measures minimizing some Lagrangian functionals and on the symplectic field theory devised by A. Floer and others [3-8,12,15] for investigating symplectic actions and Lagrangian submanifold intersections, an analog of Mather’s 𝛽-function is constructed subject to a Hamiltonian flow reduced invariantly upon some compact neighborhood of a Lagrangian submanifold. Some results on stable and unstable manifolds to hyperbolic periodic orbits having applications in the theory of adiabatic invariants of slowly perturbed integrable Hamiltonian systems are stated within the Gromov-Salamon-Zehnder [3,5,12] elliptic techniques in symplectic geometry.