F. González-Gascón

Newtonian systems of differential equations, integrable via quadratures, with trivial group of point symmetries

Abstract Examples are given of systems of second order ordinary differential equations integrable via quadratures with trivial symmetry group of local point transformations.

Ordered behaviour in force-free magnetic fields

Conditions in order that the trajectories of a force-free vectorfield lie on the level sets of a given function are studied. Force-free vectorfields symmetric under translations, rotations and roto-translations are also considered.

On the first integrals of Lotka–Volterra systems

Abstract A new method for obtaining time independent first integrals of Lotka–Volterra systems is given. By applying this method new integrable cases are found.

The inverse problem concerning symmetries of ordinary differential equations

It is shown that for any local Lie group G of transformations in R×Rn there exist differential systems of the form x(m=f(t,x,...,x(m−1), which are symmetrical under G. The order m_ of these systems is related to r_, the number of essential parameters of G.

Note on paper of Andrey concerning non-hamiltonian systems

Abstract An example is given of a dynamical system of non-zero divergence and hamiltonian. A geometrical condition is given ensuring that a dynamical system satisfying it is non-hamiltonian and an example satisfying this geometrical condition is also given.

On scattering trajectories of dynamical systems

Dynamical systems on Rn presenting geometric chaos, i.e., open domains where bounded and unbounded orbits are intermingled, have been constructed. The opposite situation (open scattering) has been studied for integrable Hamiltonian and non-Hamiltonian vector fields and for equations of type x=−∇V(x) when the potential has the form V=a(r)+b(r)F(θ) in spherical coordinates.

Limit velocity of charged particles in a constant electromagnetic field under friction

Abstract It is shown that when a particle moves in a constant electromagnetic field and is subjected to frictional forces its velocity to ds to a limit independent of the initial conditions. A limit velocity exists as well for the motion of relativistic particles in constant electric fields.

Non-wandering points of vector fields and invariant sets of functions

Abstract It is shown that the existence of a set of functions invariant under the flow of a vector field X is useful in order to preclude the existence of non-wandering points of X (fixed points, periodic orbits, orbits dense in tori, etc.).

Note on a paper of Abarbanel and Rouhi concerning Hamiltonian structures for smooth vector fields

Abstract This note gives a quick summary of why it is possible to make any vector-field, near a nonsingular point, locally hamiltonian. The method presented is constructive and considerably simpler than the one given in a recent paper by Abarbabel and Rouhi.

Note on a paper by Anderson

An error concerning the group of symmetries of a particle moving in a uniform magnetic field is corrected.