On the dynamics of mechanical systems with the homogeneous polynomial potential \(V = ax ^{4} + cx ^{2} y ^{2}\)
The authors consider a planar mechanical system \(\ddot{q} =\nabla V(q)\), \(q = (x, y) \in \mathbb R^{2}\), where the potential \(V\) is a homogeneous polynomial of degree four in two variables, \(V(x, y) = ax^{4} + cx^{2}y^{2}\), \(a, c\in \mathbb R\). The associated Hamiltonian system is \(\dot{x} = p_1\), \(\dot{y}=p_2\), with \(\dot{p_1} = 2x[2ax^{2} + cy^{2}]\) and \(\dot{p_2} = 2cx^{2}y\). The authors give relevant information about the dynamics of the Hamiltonian system as a function of the parameters \(a, c\in \mathbb R\). Also they study the Hill region as a function of the energy level and the parameters \(a\) and \(c\). The authors characterize explicitly some particular solutions of the mechanical system, namely equilibria and homothetic solutions. In general the paper is devoted to analyze certain dynamical aspects of the planar mechanical system associated to the potential \(V\).
scientific article; zbMATH DE number 5615002
scientific article