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

Abstract

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

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