Here is a (partial) answer to your question for the so-called natural Hamiltonian systems. The latter are defined as follows. Let $(M,g)$ be a (pseudo-)Riemannian manifold, $q^i$ be local coordinates on $M$ and $(q^i, p_i)$ be the adapted coordinates on $T^*M$ which is a symplectic manifold with a canonical symplectic structure as described in the question. In these coordinates the natural Hamiltonian is one of the form $$ H=\frac12 \sum\limits_{i,j=1}^n g^{ij}(q^1,\dots,q^n) p_i p_j + V(q^1,\dots,q^n), $$ where $n=\dim M$. Using the Legendre transformation we can pass to an equivalent Lagrangian $$ L=\frac12 \sum\limits_{i,j=1}^n g_{ij}(q^1,\dots,q^n) \displaystyle \frac{dq^i}{dt} \frac{dq^j}{dt}-V(q^1,\dots,q^n). $$
For such Lagrangians the answer to your question is in the affirmative if we restrict ourselves to the motion with a fixed energy $E$, i.e., on a hypersurface $H=E$. The respective solutions of the Euler--Lagrange equations associated with $L$ indeed can be viewed (see e.g. the paper Geometry of spaces with the Jacobi metric by Szydlowski, Heller and Sasin and references therein for details) as geodesics of the so-called Jacobi metric $$ h_{ij}=(E-V)g_{ij\tilde g_{ij}=(E-V)g_{ij} $$ which arises from the Maupertuis principle.
Note that here one considers the curves being the solutions of the Euler--Lagrange equations and the geodesics of $H$ \tilde g$ as (roughly speaking) one-dimensional submanifolds in $M$ ignoring their parametrization. The metric $h$ \tilde g$ obviously vanishes in the points where $E=V$, so it is in fact a degenerate metric, see e.g. the above paper by Szydlowski et al. for details.
