  We introduce a new critical value $c_\infty(L)$ for Tonelli Lagrangians $L$
on the tangent bundle of the 2-sphere without minimizing measures supported on
a point. We show that $c_\infty(L)$ is strictly larger than the Ma\~n\'e
critical value $c(L)$, and on every energy level $e\in(c(L),c_\infty(L))$ there
exist infinitely many periodic orbits of the Lagrangian system of $L$, one of
which is a local minimizer of the free-period action functional. This has
applications to Finsler metrics of Randers type on the 2-sphere. We show that,
under a suitable criticality assumption on a given Randers metric, after
rescaling its magnetic part with a sufficiently large multiplicative constant,
the new metric admits infinitely many closed geodesics, one of which is a
waist. Examples of critical Randers metrics include the celebrated Katok
metric.
