Homoclinic and Heteroclinic Orbits for a Class of Singular Planar Newtonian Systems

Abstract

The study of existence and multiplicity of solutions of differential equations possessing a variational nature is a problem of great meaning since most of them derives from mechanics and physics. In particular, this relates to Hamiltonian systems including Newtonian ones. During the past 30 years there has been a great deal of progress in the use of variational methods to find periodic, homoclinic and heteroclinic solutions of Hamiltonian systems. Hamiltonian systems with singular potentials, i.e., potentials that become infinite at a point or a larger subset of $\mathbb{R}^{n}$, are among those of the greatest interest. Let us remark that such potentials arise in celestial mechanics. For example, the Kepler problem with

$$\displaystyle{V (q) = - \frac{1} {\vert q -\xi \vert }}$$

has a point singularity at $\xi$ ($q \in \mathbb{R}^{n}\setminus \{\xi \}$). In physics, the gradient ∇V of the gravitational potential is called a weak force.