Transversal connecting orbits of Lagrangian systems with turning points: Newton–Kantorovich method

We study a natural Lagrangian system defined on a compact Riemannian manifold with potential U(q, t) = f(t)V (q). It is assumed that the factor f(t) tends to infinity as t → ±∞ and has a unique zero. It is also supposed that a set X
_c
, at which V attains its maximum or minimum, consists of nondegenerate critical points. Using variational approach together with modified Newton-Kantorovich method, we establish the existence of infinitely many connecting trajectories for such a system and provide a condition for their transversality.