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Alessandro Portaluri

Publications and source records attributed to Alessandro Portaluri.

50 records · Page 3Linked to original sources
arXiv2012

Bifurcation results for critical points of families of functionals

Recently the first author studied the bifurcation of critical points of families of functionals on a Hilbert space, which are parametrised by a compact and orientable manifold having a non-vanishing first integral cohomology group. We improve this result in two directions: topologically and analytically. From the analytical point of view we generalise it to a broader class of functionals; from the topological point of view we allow the parameter space to be a metrisable Banach manifold. Our methods are in particular powerful if the parameter space is simply connected. As an application of our results we consider families of geodesics in (semi-) Riemannian manifolds.

math.DG↗
arXiv2011

Dynamics of the the dihedral four-body problem

Consider four point particles with equal masses in the euclidean space, subject to the following symmetry constraint: at each instant they are symmetric with respect to the dihedral group $D_2$, that is the group generated by two rotations of angle $π$ around two orthogonal axes. Under a homogeneous potential of degree $-α$ for $0<α<2$, this is a subproblem of the four-body problem, in which all orbits have zero angular momentum and the configuration space is three-dimensional. In this paper we study the flow in McGehee coordinates on the collision manifold, and discuss the qualitative behavior of orbits which reach or come close to a total collision.

math.DS↗
arXiv2011

Global dynamics of stationary, dihedral, nearly-parallel vortex filaments

The goal of this paper is to give a detailed analytical description of the global dynamics of N points interacting through the singular logarithmic potential and subject to the following symmetry constraint: at each instant they form an orbit of the dihedral group of order 2l. The main device in order to achieve our results is a technique very popular in Celestial Mechanics, usually referred to as "McGehee transformation". After performing this change of coordinates that regularizes the total collision, we study the rest-points of the flow, the invariant manifolds and we derive interesting information about the global dynamics for l=2. We observe that our problem is equivalent to studying the geometry of stationary configurations of nearly-parallel vortex filaments in three dimensions in the LIA approximation.

math.DS↗
arXiv2011

Global dynamics under a weak potential on a sphere

We give a detailed analytical description of the global dynamics of a point mass moving on a sphere under the action of a logarithmic potential. After performing a McGehee-type blow-up in order to cope with the singularity of the potential, we investigate the rest-points of the flow, the invariant (stable and unstable) manifolds and we give a complete dynamical description of the motion.

math.DS↗
arXiv2009

Morse theory for a fourth order elliptic equation with exponential nonlinearity

In this paper we compute the Leray Schauder degree for a fourth order elliptic boundary value problem with exponential nonlinearity and Navier boundary condition. This will be made by proving a Poincare'-Hopf type theorem. Moreover by using this result, together with some quantitative results about the formal set of barycenters, we are able to establish a direct and geometrically clear degree counting formula for our problem. We remark that this formula has been proven with complete different methods by Lin Wei and Wang by using blow-up type estimates.

math.AP↗
arXiv2009

On a generalized Sturm theorem

Sturm oscillation theorem for second order differential equations was generalized to systems and higher order equations with positive leading coefficient by several authors. What we propose here is a Sturm oscillation theorem for systems of even order having strongly indefinite leading coefficient.

math.CA↗
arXiv2008

Brief communication. An indefinite Sturm theory

Sturm theory for second order differential equations was generalized to systems and higher order equations with positive leading coefficient by several authors. Here we propose a Sturm oscillation theorem for indefinite systems of even order and with Dirichlet boundary conditions having strongly indefinite leading term.

math.CA↗
arXiv2008

On the dihedral n-body problem

Consider n=2l>=4 point particles with equal masses in space, subject to the following symmetry constraint: at each instant they form an orbit of the dihedral group D_l, where D_l is the group of order 2l generated by two rotations of angle pi around two secant lines in space meeting at an angle of pi/l. By adding a homogeneous gravitational (Newtonian) potential one finds a special $n$-body problem with three degrees of freedom, which is a kind of generalisation of Devaney isosceles problem, in which all orbits have zero angular momentum. In the paper we find all the central configurations and we compute the dimension of the stable/unstable manifolds.

math.DS↗
arXiv2008

Maslov index of Hamiltonian systems

The aim of this paper is to give an explicit formula in order to compute the Maslov index of the fundamental solution of a linear autonomous Hamiltonian system, in terms of the Conley-Zehnder index and the time one flow.

math.DS↗
arXiv2007

A Morse Index Theorem and bifurcation for perturbed geodesics on Semi-Riemannian Manifolds

Perturbed geodesics are trajectories of particles moving on a semi-Riemannian manifold in the presence of a potential. Our purpose here is to extend to perturbed geodesics on semi-Riemannian manifolds the well known Morse Index Theorem. When the metric is indefinite, the Morse index of the energy functional becomes infinite and hence, in order to obtain a meaningful statement, we substitute the Morse index by its relative form, given by the spectral flow of an associated family of index forms. We also introduce a new counting for conjugate points, which need not to be isolated in this context, and prove that the relative Morse index equals the total number of conjugate points. Finally we study the relation with the Maslov index of the flow induced on the Lagrangian Grassmannian.

math.DG↗
arXiv2002

Spectral Flow, Maslov Index and Bifurcation of semi-Riemannian Geodesics

We give a functional analytical proof of the equality between the Maslov index of a semi-Riemannian geodesic and the spectral flow of the path of self-adjoint Fredholm operators obtained from the index form. This fact, together with recent results on the bifurcation for critical points of strongly indefinite functionals (see P. M. Fitzpatrick, J. Pejsachowicz, L. Recht, Spectral Flow and Bifurcation of Strongly Indefinite Functionals Part I. General Theory, J. Funct. Anal. 162 (1) (1999), 52-95.) imply that each non degenerate and non null conjugate (or $P$-focal) point along a semi-Riemannian geodesic is a bifurcation point.

math.DG↗