A classical problem of existence of critical curves with fixed extremes for a Lagrangian

We explain an accurate result on existence and multiplicity of critical curves of functionals with a Lagrangian of type $L(x,\dot{x},s)= \langle\dot{x},\dot{x}\rangle/2-V(x,s)$ on a complete Riemannian manifold $\m$. Concretely, the existence of such curves is obtained when $V(x,s)$ presents a subquadratic growth with respect to $x$ or, in the quadratic case, if the arrival time $T$ of the curve satisfies $0<T< T(\lambda)$. The optimal value $T(\lambda) = \pi/\sqrt{2\lambda}$, fulfilled by the harmonic oscillator, is obtained as an application of Wirtinger's inequality. Moreover, when $\m$ is not contractible, Ljusternik-Schnirelmann theory yields the existence of infinitely many such trajectories. These ideas have some applications for example in the study of the geodesic connectedness of gravitational waves.