Periodic Solutions for the Generalized Anisotropic Lennard-Jones Hamiltonian

Abstract

Abstract. We characterize the circular periodic solutions of the generalized Lennard-Jones Hamiltonian system with two particles in R n , and we analyze what of these periodic solutions can be continued to periodic solutions of the anisotropic generalized Lennard-Jones Hamiltonian system.We also characterize the periods of antiperiodic solutions of the generalized Lennard-Jones Hamiltonian system on R 2n , and prove the existences of 0 < τ * ≤ τ * * such that this system possesses no τ /2-antiperiodic solution for all τ ∈ (0, τ * ), at least one τ /2-antiperiodic solution when τ = τ * , precisely 2 n families of τ /2-antiperiodic circular solutions when τ = τ * * , and precisely 2 n+1 families of τ /2-antiperiodic circular solutions when τ > τ * * . Each of these circular solution families is of dimension n − 1 module the S 1 -action.

Positive periodic solutions for a second-order singular differential equation with two indefinite weights

“…Attractive-repulsive singularities can be regarded as a generalized Lennard-Jones force [3,4], and they are widely used in molecular dynamics to model the interaction between atomic particles [5]. By this reason, there have been some results published on differential equations with attractive-repulsive singularities [6,7,8,9,10,11].…”

Bifurcations of quasi-periodic solutions from relative equilibria in the Lennard–Jones 2-body problem

“…Lennard-Jones N -body problem has been studied in many ways, including more general class of potential. In Llibre and Long (2015), the authors have studied the circular periodic solutions and antiperiodic solutions of the generalized problem. The classical case studied here corresponds to the case γ = 0 in Llibre and Long (2015), where the authors have proved the existence of a circle of equilibria.…”

Existence of hyperbolic motions to a class of Hamiltonians and generalized $N$-body system via a geometric approach

“…It was also called as the Manev potential when α = 2 and β = 1 (see [27,15,23]), and as the Schwarzschild potential when α = 3 and β = 1 (see [1]). (E4) For all 1 ≤ i < j ≤ N , F ij is given by the Lennard-Jones potential (see [21,22]), that is,…”