arXiv subjects

Luca Asselle

Publications and source records attributed to Luca Asselle.

At least 19 recordsLinked to original sources

arXiv2026

Relative equilibria, linear stability and electromagnetic curvature

In this paper we study the linear stability of relative equilibria in the Newtonian $n$-body problem from the viewpoint of electromagnetic systems. We first examine the effect of the ambient dimension on stability, starting from the Lagrange equilateral triangle solutions of the three-body problem in $\mathbb R^4$. We then initiate a new approach to stability based on electromagnetic curvature. In a two-dimensional model, we relate linear stability to both the Ma\~n\'e critical value and to the behavior of the zero set of the electromagnetic curvature, highlighting a change in its topology at the stability threshold. This criterion is then applied to the planar $n$-body problem: in the three-body case, we recover Routh's classical criterion, and, more generally, we obtain an instability criterion for relative equilibria whose reduced linearized dynamics splits along invariant symplectic planes. These results suggest a new geometric perspective on linear stability and on questions related to Moeckel's conjecture.

math.DS↗

arXiv2023

Non-minimality and instability of brake orbits for natural Lagrangians on Riemannian manifolds

We investigate minimality and stability of periodic brake orbits in natural Lagrangian systems on smooth Riemannian manifolds. We prove that every non-constant periodic brake orbit is not a minimizer of the fixed-time action, for any conormal boundary condition. Under an orbit-cylinder hypothesis, its Morse index strictly increases in the free-time setting. As a consequence, strongly nondegenerate brake orbits fail to be linearly stable under a dimensional condition; in dimension at least three, nondegenerate mountain-pass brake orbits are spectrally unstable when the monodromy is semisimple. The key ingredient is a local index contribution at each brake instant. Using Seifert collar coordinates near the Hill boundary, we reduce the normal dynamics to a one-dimensional model, exhibiting a degeneracy inherent to brake symmetry. We illustrate the results by explicit Morse index computations for the planar anisotropic oscillator, the planar pendulum, and the planar Kepler problem; in the Kepler case, the ejection--collision orbit is treated via cotangent-lift Levi--Civita--Lissajous regularization.

math.DS↗

arXiv2023

Zoll magnetic systems on the two-torus: a Nash-Moser construction

We construct an infinite-dimensional family of smooth integrable magnetic systems on the two-torus which are Zoll, meaning that all the unit-speed magnetic geodesics are periodic. The metric and the magnetic field of such systems are arbitrarily close to the flat metric and to a given constant magnetic field. This extends to the magnetic setting a famous result by Guillemin on the two-sphere. We characterize Zoll magnetic systems as zeros of a suitable action functional $S$, and then look for its zeros by means of a Nash-Moser implicit function theorem. This requires showing the right-invertibility of the linearized operator $\mathrm{d} S$ in a neighborhood of the flat metric and constant magnetic field, and establishing tame estimates for the right inverse. As key step we prove the invertibility of the normal operator $\mathrm{d} S\circ \mathrm{d} S^*$ which, unlike in Guillemin's case, is pseudo-differential only at the highest order. We overcome this difficulty noting that, by the asymptotic properties of Bessel functions, the lower order expansion of $\mathrm{d} S \circ \mathrm{d}S^*$ is a sum of Fourier integral operators. We then use a resolvent identity decomposition which reduces the problem to the invertibility of $\mathrm{d} S \circ \mathrm{d} S^*$ restricted to the subspace of functions corresponding to high Fourier modes. The inversion of such a restricted operator is finally achieved by making the crucial observation that lower order Fourier integral operators satisfy asymmetric tame estimates.

math.DG↗

arXiv2021

Spectral stability, spectral flow and circular relative equilibria for the Newtonian $n$-body problem

For the Newtonian (gravitational) $n$-body problem in the Euclidean $d$-dimensional space, $d\ge 2$, the simplest possible periodic solutions are provided by circular relative equilibria, (RE) for short, namely solutions in which each body rigidly rotates about the center of mass and the configuration of the whole system is constant in time and central (or, more generally, balanced) configuration. For $d\le 3$, the only possible (RE) are planar, but in dimension four it is possible to get truly four dimensional (RE). A classical problem in celestial mechanics aims at relating the (in-)stability properties of a (RE) to the index properties of the central (or, more generally, balanced) configuration generating it. In this paper, we provide sufficient conditions that imply the spectral instability of planar and non-planar (RE) in $\mathbb R^4$ generated by a central configuration, thus answering some of the questions raised in \cite[Page 63]{Moe14}. As a corollary, we retrieve a classical result of Hu and Sun \cite{HS09} on the linear instability of planar (RE) whose generating central configuration is non-degenerate and has odd Morse index, and fix a gap in the statement of \cite[Theorem 1]{BJP14} about the spectral instability of planar (RE) whose (possibly degenerate) generating central configuration has odd Morse index. The key ingredients are a new formula of independent interest that allows to compute the spectral flow of a path of symmetric matrices having degenerate starting point, and a symplectic decomposition of the phase space of the linearized Hamiltonian system along a given (RE) which is inspired by Meyer and Schmidt's planar decomposition \cite{MS05} and which allows us to rule out the uninteresting part of the dynamics corresponding to the translational and (partially) to the rotational symmetry of the problem.

math.DS↗

arXiv2021

Morse theory for $S$-balanced configurations in the Newtonian $n$-body problem

For the Newtonian (gravitational) $n$-body problem in the Euclidean $d$-dimensional space, the simplest possible solutions are provided by those rigid motions (homographic solutions) in which each body moves along a Keplerian orbit and the configuration of the $n$-body is a constant up to rotations and scalings named \textit{central configuration}. For $d\leq 3$, the only possible homographic motions are those given by central configurations. For $d \geq 4$ instead, new possibilities arise due to the higher complexity of the orthogonal group $O(d)$, as observed by Albouy and Chenciner. For instance, in $\mathbb R^4$ it is possible to rotate in two mutually orthogonal planes with different angular velocities. This produces a new balance between gravitational forces and centrifugal forces providing new periodic and quasi-periodic motions. So, for $d\geq 4$ there is a wider class of $S$-\textit{balanced configurations} (containing the central ones) providing simple solutions of the $n$-body problem, which can be characterized as well through critical point theory. In this paper, we first provide a lower bound on the number of balanced (non-central) configurations in $\mathbb R^d$, for arbitrary $d\geq 4$, and establish a version of the $45^\circ$-theorem for balanced configurations, thus answering some questions raised by Moeckel. Also, a careful study of the asymptotics of the coefficients of the Poincaré polynomial of the collision free configuration sphere will enable us to derive some rather unexpected qualitative consequences on the count of $S$-balanced configurations. In the last part of the paper, we focus on the case $d=4$ and provide a lower bound on the number of periodic and quasi-periodic motions of the gravitational $n$-body problem which improves a previous celebrated result of McCord.

math.DS↗

arXiv2021

Normal forms for strong magnetic systems on surfaces: Trapping regions and rigidity of Zoll systems

We prove a normal form for strong magnetic fields on a closed, oriented surface and use it to derive two dynamical results for the associated flow. First, we show the existence of KAM tori and trapping regions provided a natural non-resonance condition holds. Second, we prove that the flow cannot be Zoll unless (i) the Riemannian metric has constant curvature and the magnetic function is constant, or (ii) the magnetic function vanishes and the metric is Zoll. We complement the second result by exhibiting an exotic magnetic field on a flat two-torus yielding a Zoll flow for arbitrarily small rescalings.

math.DS↗

arXiv2020

Bifurcations of balanced configurations for the Newtonian $n$-body problem in $\mathbb R^4$

For the gravitational $n$-body problem, the simplest motions are provided by those rigid motions in which each body moves along a Keplerian orbit and the shape of the system is a constant (up to rotations and scalings) configuration featuring suitable properties. While in dimension $d \leq 3$ the configuration must be central, in dimension $d \geq 4$ new possibilities arise due to the complexity of the orthogonal group, and indeed there is a wider class of $S$-balanced configurations, containing central ones, which yield simple solutions of the $n$-body problem. Starting from recent results of the first and third authors, we study the existence of continua of bifurcations branching from a trivial branch of collinear $S$-balanced configurations and provide an estimate from below on the number of bifurcation instants. In the last part of the paper, by using the continuation method, we explicitly display the bifurcation branches in the case of the three body problem for different choices of the masses.

math.DS↗

arXiv2020

The Palais-Smale condition for the Hamiltonian action on a mixed regularity space of loops in cotangent bundles and applications

We show that the Hamiltonian action satisfies the Palais-Smale condition over a "mixed regularity" space of loops in cotangent bundles, namely the space of loops with regularity $H^s$, $s\in (\frac 12, 1)$, in the base and $H^{1-s}$ in the fiber direction. As an application, we give a simplified proof of a theorem of Hofer-Viterbo on the existence of closed characteristic leaves for certain contact type hypersufaces in cotangent bundles.

math.SG↗

arXiv2019

On the rigidity of Zoll magnetic systems on surfaces

In this paper we study rigidity aspects of Zoll magnetic systems on closed surfaces. We characterize magnetic systems on surfaces of positive genus given by constant curvature metrics and constant magnetic functions as the only magnetic systems such that the associated Hamiltonian flow is Zoll, i.e. every orbit is closed, on every energy level. We also prove the persistence of possibly degenerate closed geodesics under magnetic perturbations in different instances.

math.DS↗

arXiv2019

Waist theorems for Tonelli systems in higher dimensions

We study the periodic orbits problem on energy levels of Tonelli Lagrangian systems over configuration spaces of arbitrary dimension. We show that, when the fundamental group is finite and the Lagrangian has no stationary orbit at the Mañé critical energy level, there is a waist on every energy level just above the Mañé critical value. With a suitable perturbation with a potential, we show that there are infinitely many periodic orbits on every energy level just above the Mañé critical value, and on almost every energy level just below. Finally, we prove the Tonelli analogue of a closed geodesics result due to Ballmann-Thorbergsson-Ziller.

math.DS↗

arXiv2019

Integrable magnetic flows on the two-torus: Zoll examples and systolic inequalities

In this paper we study some aspects of integrable magnetic systems on the two-torus. On the one hand, we construct the first non-trivial examples with the property that all magnetic geodesics with unit speed are closed. On the other hand, we show that those integrable magnetic systems admitting a global surface of section satisfy a sharp systolic inequality.

math.DS↗

arXiv2018

Minimal boundaries in Tonelli Lagrangian systems

We prove several new results concerning action minimizing periodic orbits of Tonelli Lagrangian systems on an oriented closed surface $M$. More specifically, we show that for every energy larger than the maximal energy of a constant orbit and smaller than or equal to the Mañé critical value of the universal abelian cover, the Lagrangian system admits a minimal boundary, i.e. a global minimizer of the Lagrangian action on the space of smooth boundaries of open sets of $M$. We also extend the celebrated graph theorem of Mather in this context: in the tangent bundle $TM$, the union of the supports of all lifted minimal boundaries with a given energy projects injectively to the base $M$. Finally, we prove the existence of action minimizing simple periodic orbits on energies just above the Mañé critical value of the universal abelian cover. This provides in particular a class of non-reversible Finsler metrics on the 2-sphere possessing infinitely many closed geodesics.

math.DS↗

arXiv2018

On Tonelli periodic orbits with low energy on surfaces

We prove that, on a closed surface, a Lagrangian system defined by a Tonelli Lagrangian $L$ possesses a periodic orbit that is a local minimizer of the free-period action functional on every energy level belonging to the low range of energies $(e_0(L),c_{\mathrm{u}}(L))$. We also prove that almost every energy level in $(e_0(L),c_{\mathrm{u}}(L))$ possesses infinitely many periodic orbits. These statements extend two results, respectively due to Taimanov and Abbondandolo-Macarini-Mazzucchelli-Paternain, valid for the special case of electromagnetic Lagrangians.

math.DS↗

arXiv2017

Closed geodesics with local homology in maximal degree on non-compact manifolds

We show that, on a complete and possibly non-compact Riemannian manifold of dimension at least 2 without close conjugate points at infinity, the existence of a closed geodesic with local homology in maximal degree and maximal index growth under iteration forces the existence of infinitely many closed geodesics. For closed manifolds, this was a theorem due to Hingston.

math.DG↗

arXiv2017

On geodesic flows with symmetries and closed magnetic geodesics on orbifolds

Let $Q$ be a closed manifold admitting a locally-free action of a compact Lie group $G$. In this paper we study the properties of geodesic flows on $Q$ given by Riemannian metrics which are invariant by such an action. In particular, we will be interested in the existence of geodesics which are closed up to the action of some element in the group $G$, since they project to closed magnetic geodesics on the quotient orbifold $Q/G$.

math.DS↗

arXiv2017

On the existence of closed magnetic geodesics via symplectic reduction

Let $(M,g)$ be a closed Riemannian manifold and $σ$ be a closed 2-form on $M$ representing an integer cohomology class. In this paper, using symplectic reduction, we show how the problem of existence of closed magnetic geodesics for the magnetic flow of the pair $(g,σ)$ can be interpreted as a critical point problem for a Rabinowitz-type action functional defined on the cotangent bundle $T^*E$ of a suitable $S^1$-bundle $E$ over $M$ or, equivalently, as a critical point problem for a Lagrangian-type action functional defined on the free loopspace of $E$. We then study the relation between the stability property of energy hypersurfaces in $(T^*M,dp\wedge dq+π^*σ)$ and of the corresponding codimension 2 coisotropic submanifolds in $(T^*E,dp\wedge dq)$ arising via symplectic reduction. Finally, we reprove the main result of [9] in this setting.

math.DS↗

arXiv2016

On the existence of infinitely many closed geodesics on non-compact manifolds

We prove that any complete (and possibly non-compact) Riemannian manifold $M$ possesses infinitely many closed geodesics provided its free loop space has unbounded Betti numbers in degrees larger than the dimension of $M$, and there are no close conjugate points at infinity. Our argument builds on an existence result due to Benci and Giannoni, and generalizes the celebrated theorem of Gromoll and Meyer for closed manifolds.

math.DG↗

arXiv2016

On the periodic motions of a charged particle in an oscillating magnetic field on the two-torus

Let $(\mathbb T^2,g)$ be a Riemannian two-torus and let $σ$ be an oscillating $2$-form on $\mathbb T^2$. We show that for almost every small positive number $k$ the magnetic flow of the pair $(g,σ)$ has infinitely many periodic orbits with energy $k$. This result complements the analogous statement for closed surfaces of genus at least $2$ [Asselle and Benedetti, Calc. Var. Partial Differential Equations, 2015] and at the same time extends the main theorem in [Abbondandolo, Macarini, Mazzucchelli, and Paternain, J. Eur. Math. Soc. (JEMS), to appear] to the non-exact oscillating case.

math.DS↗