Claudio Vidal
Av. Prof. Luiz Freire, s/n, Cidade Universitaria, RecifePe, Brasil
Departamento de Matematica, Universidade Federal de Pernambuco
Publications:
Schmidt D. S., Vidal C.
Stability of the Planar Equilibrium Solutions of a Restricted $1+N$ Body Problem
2014, vol. 19, no. 5, pp. 533547
Abstract
We started our studies with a planar Eulerian restricted fourbody problem (ERFBP) where three masses move in circular orbits such that their configuration is always collinear. The fourth mass is small and does not influence the motion of the three primaries. In our model we assume that one of the primaries has mass 1 and is located at the origin and two masses of size $\mu$ rotate around it uniformly. The problem was studied in [3], where it was shown that there exist noncollinear equilibria, which are Lyapunov stable for small values of $\mu$. KAM theory is used to establish the stability of the equilibria. Our computations do not agree with those given in [3], although our conclusions are similar. The ERFBP is a special case of the $1+N$ restricted body problem with $N=2$. We are able to do the computations for any $N$ and find that the stability results are very similar to those for $N=2$. Since the $1+N$ body configuration can be stable when $N>6$, these results could be of more significance than for the case $N=2$.

Dos Santos F., Vidal C.
Stability of Equilibrium Solutions of Hamiltonian Systems Under the Presence of a Single Resonance in the NonDiagonalizable Case
2008, vol. 13, no. 3, pp. 166177
Abstract
The problem of knowing the stability of one equilibrium solution of an analytic autonomous Hamiltonian system in a neighborhood of the equilibrium point in the case where all eigenvalues are pure imaginary and the matrix of the linearized system is nondiagonalizable is considered.We give information about the stability of the equilibrium solution of Hamiltonian systems with two degrees of freedom in the critical case. We make a partial generalization of the results to Hamiltonian systems with $n$ degrees of freedom, in particular, this generalization includes those in [1].

Santos A., Vidal C.
Symmetry of the Restricted 4+1 Body Problem with Equal Masses
2007, vol. 12, no. 1, pp. 2738
Abstract
We consider the problem of symmetry of the central configurations in the restricted 4+1 body problem when the four positive masses are equal and disposed in symmetric configurations, namely, on a line, at the vertices of a square, at the vertices of a equilateral triangle with a mass at the barycenter, and finally, at the vertices of a regular tetrahedron [13]. In these situations, we show that in order to form a non collinear central configuration of the restricted 4+1 body problem, the null mass must be on an axis of symmetry. In our approach, we will use as the main tool the quadratic forms introduced by A. Albouy and A. Chenciner [4]. Our arguments are general enough, so that we can consider the generalized Newtonian potential and even the logarithmic case. To get our results, we identify some properties of the Newtonian potential (in fact, of the function $\varphi (s)=s^k$, with $k<0$) which are crucial in the proof of the symmetry.

Vidal C., Dos Santos F.
Stability of equilibrium positions of periodic Hamiltonian systems under third and fourth order resonances
2005, vol. 10, no. 1, pp. 95111
Abstract
The problem of the stability of an equilibrium position of a nonautonomous 2$\pi$periodic Hamiltonian system with $n$ degrees of freedom ($n \geqslant 2$), in a nonlinear setting, is studied in the presence of a single third and fourth order resonance. We give conditions of instability in the sense of Lyapunov and formal stability of the equilibrium position, depending on the coefficients of the Hamiltonian function.
