38 research outputs found
Symbolic dynamics for the -centre problem at negative energies
We consider the planar -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 the centres inside a compact set. The proof is based upon
topological, variational and geometric arguments. The existence result allows
to characterize the associated dynamical system with a symbolic dynamics, where
the symbols are the partitions of the centres in two non-empty sets
Simple choreographies of the planar Newtonian -body Problem
In the -body problem, a simple choreography is a periodic solution, where
all masses chase each other on a single loop. In this paper we prove that for
the planar Newtonian -body problem with equal masses, , there are
at least different main simple choreographies. This
confirms a conjecture given by Chenciner and etc. in \cite{CGMS02}.Comment: 31pages, 6 figures. Refinements in notations and proof
Action minimizing orbits in the n-body problem with simple choreography constraint
In 1999 Chenciner and Montgomery found a remarkably simple choreographic
motion for the planar 3-body problem (see \cite{CM}). In this solution 3 equal
masses travel on a eight shaped planar curve; this orbit is obtained minimizing
the action integral on the set of simple planar choreographies with some
special symmetry constraints. In this work our aim is to study the problem of
masses moving in \RR^d under an attractive force generated by a potential
of the kind , , with the only constraint to be a simple
choreography: if are the orbits then we impose the
existence of x \in H^1_{2 \pi}(\RR,\RR^d) such that q_i(t)=x(t+(i-1) \tau),
i=1,...,n, t \in \RR, where . In this setting, we first
prove that for every d,n \in \NN and , the lagrangian action
attains its absolute minimum on the planar circle. Next we deal with the
problem in a rotating frame and we show a reacher phenomenology: indeed while
for some values of the angular velocity minimizers are still circles, for
others the minima of the action are not anymore rigid motions.Comment: 24 pages; 4 figures; submitted to Nonlinearit
On the minimality of Keplerian arcs with fixed negative energy
We revisit a classical result by Jacobi on the local minimality, as critical
points of the corresponding energy functional, of fixed-energy solutions of the
Kepler equation joining two distinct points with the same distance from the
origin. Our proof relies on the Morse index theorem, together with a
characterization of the conjugate points as points of geodesic bifurcation
A note on the radial solutions for the supercritical Hénon equation
We prove the existence of a positive radial solution for the Hénon equation with arbitrary growth. The solution is found by means
of a shooting method and turns out to be an increasing function of the radial variable. Some numerical experiments suggest the
existence of many positive oscillating solutions
Existence and multiplicity of positive solutions for a Dirichlet boundary value problem in R-2
We deal with the Dirichlet boundary value problem associated to a parameter-dependent second order vector differential equation. Using the method of lower and upper solutions together with degree theory, we provide existence and multiplicity of positive solutions