  We revisit the existence problem of heteroclinic connections in
$\mathbb{R}^N$ associated with Hamiltonian systems involving potentials
$W:\mathbb{R}^N\to \mathbb{R}$ having several global minima. Under very mild
assumptions on $W$ we present a simple variational approach to first find
geodesics minimizing length of curves joining any two of the potential wells,
where length is computed with respect to a degenerate metric having conformal
factor $\sqrt{W}.$ Then we show that when such a minimizing geodesic avoids
passing through other wells of the potential at intermediate times, it gives
rise to a heteroclinic connection between the two wells. This work improves
upon the approach of P.Sternberg in $\texttt{Vector-valued local minimizers of
nonconvex}$ $\texttt{variational problems}$, and represents a more geometric
alternative to the approaches for finding such connections described, for
example, by N.D. Alikakos and G.Fusco in $\texttt{On the connection problem for
potentials with}$ $\texttt{several global minima}$, by S.V. Bolotin in
$\texttt{Libration motions of natural dynamical systems}$, by J. Byeon, P.
Montecchiari, and P. Rabinowitz in $\texttt{A double well potential}$
$\texttt{system}$, and by P. Rabinowitz in $\texttt{Homoclinic and heteroclinic
orbits for a class of Hamiltonian}$ $\texttt{systems}$.
