Non-collision Orbits for a Class of Singular Hamiltonian Systems on the Plane with Weak Force Potentials

We study the existence of non-collision orbits for a class of singular Hamiltonian systems

$$\begin{aligned} \ddot{q}+ V'(q)=0 \end{aligned}$$

where \(q:{\mathbb {R}} \longrightarrow {\mathbb {R}}^2\) and \(V\in C^2({\mathbb {R}}^2 {\setminus } \{e\},\, {\mathbb {R}})\) is a potential with a singularity at a point \(e\not =0\). We consider V which behaves like \(\displaystyle -1/|q-e|^\alpha \) as \( q\rightarrow e \) with \(\alpha \in ]0,2[.\) Under the assumption that 0 is a strict global maximum for V, we establish the existence of a homoclinic orbit emanating from 0. Moreover, in case \(\displaystyle V(q) \longrightarrow 0\) as \(|q|\rightarrow +\infty \), we prove the existence of a heteroclinic orbit “at infinity" i.e. a solution q such that

$$\begin{aligned} \lim _{t\rightarrow -\infty } q(t)=0,\,\, \lim _{t \rightarrow +\infty }|q(t)|=+\infty \,\, \hbox {and} \, \lim _{t \rightarrow \pm \infty }\dot{q}(t)=0. \end{aligned}$$