相关论文: Periodic Solutions of the Planar N-Center Problem …

In a singular potential setting, we generalize a method which allows to show that minimizers under topological constraints of the action functional (or of the Maupertuis') are collision-free. This methods applies to $3$-dimensional problems…

We consider the planar $N$-centre problem, with homogeneous potentials of degree $-\a<0$, $\a \in [1,2)$. We prove the existence of infinitely many collisions-free periodic solutions with negative and small energy, for any distribution of…

We study the existence of non-collision periodic solutions with Newtonian potentials for the following planar restricted 4-body problems: Assume that the given positive masses $m_{1},m_{2},m_{3}$ in a Lagrange configuration move in circular…

We investigate the existence of collision-free nonconstant periodic solutions of the $N$-vortex problem in domains $\Omega\subset\mathbb{C}$. These are solutions $z(t)=(z_1(t),\dots,z_N(t))$ of the first order Hamiltonian system \[…

We use variational minimizing methods to study spatial restricted N+1-body problems with a zero mass moving on the vertical axis of the moving plane for N equal masses. We prove that the minimizer of the Lagrangian action on the anti-T/2 or…

The family of planar linear chains are found as collision-free action minimizers of the spatial $N$-body problem with equal masses under $D_N$ or $D_N \times \zz_2$-symmetry constraint and different types of topological constraints. This…

We consider a Kepler problem in dimension two or three, with a time-dependent $T$-periodic perturbation. We prove that for any prescribed positive integer $N$, there exist at least $N$ periodic solutions (with period $T$) as long as the…

In this paper, we prove the existence of a family of new non-collision periodic solutions for the classical Newtonian $n$-body problems. In our assumption, the $n=2l\geq4$ particles are invariant under the dihedral rotation group $D_l$ in…

In this paper we consider a class of planar autonomous systems having an isolated limit cycle x_0 of smallest period T>0 such that the associated linearized system around it has only one characteristic multiplier with absolute value 1. We…

In this paper, we study the existence of non-planar periodic solutions for the following spatial restricted 3-body and 4-body problems: for $N=2 or 3$, given any masses $m_{1},...,m_{N}$, the mass points of $m_{1},...,m_{N}$ move on the $N$…

Aim of this paper is to investigate the existence of periodic solutions of a nonlinear planar autonomous system having a limit cycle x_0 of least period T_0>0 when it is perturbed by a small parameter, T_1-periodic, perturbation. In the…

In this work we prove the lower bound for the number of $T$-periodic solutions of an asymptotically linear planar Hamiltonian system. Precisely, we show that such a system, $T$-periodic in time, with $T$-Maslov indices $i_0,i_\infty$ at the…

We study relativistic Kepler problems in the plane. At first, using non-smooth critical point theory, we show that under a general time-periodic external force of gradient type there are two infinite families of T-periodic solutions,…

The configuration space of the planar three-body problem when collisions are excluded has a rich topology which supports a large set of free homotopy classes. Most classes survive modding out by rotations. Those that survive are called the…

We consider the $N$-body problem in $\mathbb{R}^d$ with the newtonian potential $1/r$. We prove that for every initial configuration $x_i$ and for every minimizing normalized central configuration $x_0$, there exists a collision-free…

We continue the variational approach to parabolic trajectories introduced in our previous paper [5], which sees parabolic orbits as minimal phase transitions. We deepen and complete the analysis in the planar case for homogeneous singular…

In this paper, we apply the variational method with the Structural Prescribed Boundary Conditions (SPBC) to prove the existence of periodic and quasi-periodic solutions for planar four-body problem with $m_1=m_3$ and $m_2=m_4$. A path…

An action minimizing path between two given configurations, spatial or planar, of the $n$-body problem is always a true -- collision-free -- solution. Based on a remarkable idea of Christian Marchal, this theorem implies the existence of…

It is shown that in the planar equal-mass four-body problem, there exist two sets of new action minimizers connecting two planar boundary configurations with fixed symmetry axes and specific order constraints on the four bodies: a double…

Periodic and quasi-periodic solutions of the n-body problem can be found as minimizers of the Lagrangian action functional restricted to suitable spaces of symmetric paths. The main purpose of this paper is to develop a systematic approach…