If $M$ is a locally convex compact manifold (possibly with boundary), $E$ also satisfies the Palais-Smale condition and the $(-\nabla E)$-flow is defined for all time $t\geq 0$. We notice that every closed geodesic lies on a critical circle $S^1\cdot\gamma$, where $S^1$ acts on $\Lambda M$ by $t\cdot\gamma :=
\gamma(t+\cdot)$. In our study we assume that $E$ has only isolated critical circles (except for the constant loops which have zero value). Two closed geodesics $c_1$ and $c_2$ are said to be geometrically distinct if they do not have the same image in $M$.

Finite-dimensional approximation of the loop space
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Morse’s finite-dimensional approximation of the curve space over $M$, as presented by Bangert in [@Ban80] consist of the following data: an open set $\mathcal{O}\subset M$, an energy bound $\kappa>0$ and a parameter $j\in{{\mathbb{N}}}$ satisfying $\frac{1}{j} <
\frac{\varepsilon^2}{\kappa}$ where $\varepsilon > 0$ is smaller than the injectivity radius on $\mathcal{O}$. The positivity of $\varepsilon$ will be fulfilled if for instance $\mathcal{O}$ has compact closure, as will be the case in our considerations. The finite-dimensional approximation $\Omega=\Omega(\mathcal{O},\kappa,j)$ is constructed as follows: it is the set of all curves $\gamma\in\Lambda M$ such that $E(\gamma)<\kappa$, $\gamma(i/j)\in\mathcal{O}$ and such that $\gamma|_{[i/j,(i+1)/j]}$ is a geodesic of 