Calculus of variations in the large, existence of trajectories in a domain with boundary, and Whitney's inverted pendulum problem

For non-autonomous Lagrangian systems we introduce the notion of a dynamically convex domain with respect to the Lagrangian. We establish the solubility of boundary-value problems in compact dynamically convex domains. If the Lagrangian is time-periodic, then such a domain contains a periodic trajectory. The proofs use the Hamilton principle and known tools of the calculus of variations in the large. Our general results are applied to Whitney's problem on the existence of motions of an inverted pendulum without falls.