In lines 8-11 of \cite[pp. 2977]{Lu} (arXiv:0806.0425 [math.SG], 3 Jun 2008)
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, it was
recently proved in \cite{AbSc1} (arXiv:0812.4364 [math.DS], 23 Dec 2008) that
the claim of $C^2$-smoothness of ${\cal L}_\tau$ in \cite[Prop.4.1]{AbF} is
only true if for every $(t,q)$ the function $v\mapsto L(t,q,v)$ is a polynomial
of degree at most 2, and that the functional ${\cal L}_\tau$ is only $C^{2-0}$
under the assumptions (L1)-(L3). So the arguments in \cite{Lu} is only valid
for the physical Hamiltonian in (1.2) and corresponding Lagrangian therein.
Fortunately, the generalizations of Morse lemma in \cite{JM} or \cite{LLL} can
be used to overcome this defect. In this note we shall correct our arguments in
\cite{Lu} with a generalized Morse lemma in \cite{JM}. There also exists
another method to overcome this difficulty, see \cite{Ma}(arXiv:0810.2108
[math.DS], 12 Oct 2008).