  In lines 8-11 of \cite[pp. 2977]{Lu} we wrote: "For integer $m\ge 3$, if $M$
is $C^m$-smooth and
  $C^{m-1}$-smooth $L:\R\times TM\to\R$ satisfies the assumptions (L1)-(L3),
then the functional ${\cal L}_\tau$ is $C^2$-smooth, bounded below, satisfies
the Palais-Smale condition, and all critical points of it have finite Morse
indexes and nullities (see \cite[Prop.4.1, 4.2]{AbF} and \cite{Be})." However,
as proved in \cite{AbSc1} the claim that ${\cal L}_\tau$ is $C^2$-smooth is
true if and only if for every $(t,q)$ the function $v\mapsto L(t,q,v)$ is a
polynomial of degree at most 2. So the arguments in \cite{Lu} is only valid for
the physical Hamiltonian in (1.2) and corresponding Lagrangian therein. In this
note we shall correct our arguments in \cite{Lu} with a new splitting lemma
obtained in \cite{Lu2}.
