Heteroclinic Orbits for Singular Hamiltonian Systems

Abstract

We are concerned with the existence of heteroclinic orbits for singular Hamiltonian systems of second order $\ddot{q}(t) + \nabla V(t, \,q)=0 $ where $V(t,\,q)$ is periodic in $t$ and has a singularity at a~point~${q=e}$. Suppose $V$ possesses a global maximum $\overline V$ on $\mathbb R \times \mathbb R ^N\setminus\{e\}$ and $V(t,\,x)= \overline{V}$ if and only if $x\in \mathcal{M}$ where $\mathcal{M}$ contains at least two points and consists only of isolated points. Under these and suitable conditions on $V$ near $q=e$ and at infinity, we show for each $a_0^{}\in \mathcal M$, the existence of at least one heteroclinic orbit joining $a_0^{}$ to $\mathcal M \setminus\{a_0^{}\}$. Two different settings are studied. For the first, the usual strong force condition of Gordon near the singularity is assumed. For the second, the potential $V$ behaves near $q=e$ like $-\frac1{|q-e|^\alpha}$ with $0<\alpha<2$ (the weak force case). In both cases the existence of heteroclinic orbits $q\colon\mathbb R \to\mathbb R^N\setminus\{e\}$ is obtained via a~minimization of the corresponding action functional.