Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds (Q632356)

The article deals with the following Hamiltonian system \[ \begin{aligned} \dot q_i &= \frac{\partial H}{\partial p_i} = g^{ij}(q)p_j, \\ \dot p_i &= - \frac{\partial H}{\partial q_i} = -\frac12 \frac{\partial g^{kl}}{\partial q_i} p_kp_l - \frac{\partial V}{\partial q_i}, \end{aligned} \qquad i = 1,\dots,n \] on a smooth Riemannian manifold \({\mathcal M}\) with the metric \((g_{ij})\) of dimension \(n\), \(H = \frac12 g^{ij}(q)p_ip_j + V(t,q)\), \((g^{ij})\) is the inverse matrix of \((g_{ij})\). It is assumed that: (\(a\)) \(V \in C^1({\mathbb R} \times {\mathbb R}^n,{\mathbb R})\), \(V \leq 0\) on \({\mathbb R} \times {\mathbb R}^n\); (\(b\)) for \({\mathcal V} = \{q \in {\mathcal M}:\;V(t,q) = 0 \;\text{for all} \;t \in {\mathbb R}\}\) we have \(\# {\mathcal V} \geq 2\) and \(\sigma = \frac13 \min \;\{\rho(x,y):\;x, y \in {\mathcal V}, x \neq y\} > 0\) and for \(q \notin {\mathcal V}\) \(V(t,q) \neq 0\) for any \(t \in {\mathbb R}\); (\(c\)) in any compact subset of \({\mathcal M} \setminus {\mathcal V}\) we have \(\int_0^\infty V(t,q) \, dt = \int_{-\infty}^0 V(t,q) \, dt = - \infty\); and either, (\(d_1\)) in any compact subset of \({\mathcal M} \setminus {\mathcal V}\) we have \(-V(t,q) \to \infty\) if \(|t| \to \infty\); (\(d_2\)) for any \(\varepsilon > 0\) there exists a \(\delta > 0\) such that \(q \in B_\delta({\mathcal V})\) implies \(-V(t,q) \leq \varepsilon\). The authors prove that, under these conditions, for any \(x \in {\mathcal V}\) there exists \(y \in {\mathcal V} \setminus \{x\}\) for which the system under consideration has a heteroclinic orbit connected \(x\) and \(y\), or, in other words, there exists a complete solution \(q(t)\) of the system satisfying \(q(-\infty) x\) and \(q(+\infty) = y\).