It is known that for $n \geq 3$ centres and positive energies the $n$-centre problem of celestial mechanics leads to a flow with a strange repellor and positive topological entropy.
Here we consider the energies above some threshold and show: Whereas for arbitrary $g >1$ independent integrals of Gevrey class $g$ exist, no real-analytic (that is, Gevrey class 1) independent integral exists.
We prove that the minimum-time controlled Kepler problem is not Liouville integrable in the class of meromorphic functions, via the Moralès-Ramis theory.
We solve the analytic integrability problem for diferential systems in the plane whose origin is an isolated singularity and the first homogeneous component is a quadratic Lotka-Volterra type. As an application, we give the analytically integrable systems of a class of systems $\dot{x}= x(P1+P2)$; $\dot{y}= y(Q1+Q2)$; being Pi,Qi homogeneous polynomials of degree i.
We present a new family of integrable versions of the Euler two-centre problem on two-dimensional sphere in the presence of the Dirac magnetic monopole of arbitrary charge. The new systems have very special algebraic potential and additional integral quadratic in momenta, both in classical and quantum versions.
In this work it is solved the analytic integrability problem around a nilpotent singularity of a differential system in the plane under generic conditions.
In this paper we use the orbital normal form of the nondegenerate Hopf-zero singularity to obtain necessary conditions for the existence of first integrals for such singularity. Also, we analyze the relation between the existence of first integrals and of inverse Jacobi multipliers. Some algorithmic procedures for determining the existence of first integrals are presented, and they are applied to some families of vector fields.
The main objective of this work is to investigate the integrability and linearizability problems around a singular point at the origin of the family of differential systems Particularly we are interested in the three-dimensional cubic Lotka-Volterra systems for (1:-1:1) resonant. We give a complete set of necessary conditions for both integrability and linearizability problems. The local existences of two independent first integrals guarantee that these conditions are sufficient as well. Furthermore, we use the Darboux method of integrability and some other techniques to show that the system a…
The planar $N$-centre problem describes the motion of a particle moving in the plane under the action of the force fields of $N$ fixed attractive centres: \[ \ddot{x}(t)=\sum_{j=1}^N\nabla V_j(x-c_j). \] In this paper we prove symbolic dynamics at slightly negative energy for an $N$-centre problem where the potentials $V_j$ are positive, anisotropic and homogeneous of degree $-α_j$: \[ V_j(x)=|x|^{-α_j}V_j\left(\frac{x}{|x|}\right). \] The proof is based on a broken geodesics argument and trajectories are extremals of the Maupertuis' functional. Compared with the classical $N$-centre problem w…
