Existence of periodic orbits near heteroclinic connections

We consider a potential $W:R^m\rightarrow R$ with two different global minima $a_-, a_+$ and, under a symmetry assumption, we use a variational approach to show that the Hamiltonian system \begin{equation} \ddot{u}=W_u(u), \hskip 2cm (1) \end{equation} has a family of $T$-periodic solutions $u^T$ which, along a sequence $T_j\rightarrow+\infty$, converges locally to a heteroclinic solution that connects $a_-$ to $a_+$. We then focus on the elliptic system \begin{equation} \Delta u=W_u(u),\;\; u:R^2\rightarrow R^m, \hskip 2cm (2) \end{equation} that we interpret as an infinite dimensional analogous of (1), where $x$ plays the role of time and $W$ is replaced by the action functional $$J_R(u)=\int_R\Bigl(\frac{1}{2}\vert u_y\vert^2+W(u)\Bigr)dy.$$ We assume that $J_R$ has two different global minimizers $\bar{u}_-, \bar{u}_+:R\rightarrow R^m$ in the set of maps that connect $a_-$ to $a_+$. We work in a symmetric context and prove, via a minimization procedure, that (2) has a family of solutions $u^L:R^2\rightarrow R^m$, which is $L$-periodic in $x$, converges to $a_\pm$ as $y\rightarrow\pm\infty$ and, along a sequence $L_j\rightarrow+\infty$, converges locally to a heteroclinic solution that connects $\bar{u}_-$ to $\bar{u}_+$.Comment: 36 pages, 4 figure

On the stability of periodic N-body motions with the symmetry of Platonic polyhedra

In (Fusco et. al., 2011) several periodic orbits of the Newtonian N-body problem have been found as minimizers of the Lagrangian action in suitable sets of T-periodic loops, for a given T>0. Each of them share the symmetry of one Platonic polyhedron. In this paper we first present an algorithm to enumerate all the orbits that can be found following the proof in (Fusco et. al., 2011). Then we describe a procedure aimed to compute them and study their stability. Our computations suggest that all these periodic orbits are unstable. For some cases we produce a computer-assisted proof of their instability using multiple precision interval arithmetic

On the nodal distance between two Keplerian trajectories with a common focus

We study the possible values of the nodal distance $\delta_{\rm nod}$ between two non-coplanar Keplerian trajectories ${\cal A}, {\cal A}'$ with a common focus. In particular, given ${\cal A}'$ and assuming it is bounded, we compute optimal lower and upper bounds for $\delta_{\rm nod}$ as functions of a selected pair of orbital elements of ${\cal A}$, when the other elements vary. This work arises in the attempt to extend to the elliptic case the optimal estimates for the orbit distance given in (Gronchi and Valsecchi 2013) in case of a circular trajectory ${\cal A}'$. These estimates are relevant to understand the observability of celestial bodies moving (approximately) along ${\cal A}$ when the observer trajectory is (close to) ${\cal A}'$.Comment: 34 pages, 34 figure

Platonic polyhedra, periodic orbits and chaotic motions in the N-body problem with non-Newtonian forces

We consider the $N$-body problem with interaction potential $U_alpha=rac{1}{ert x_i-x_jert^alpha}$ for alpha>1. We assume that the particles have all the same mass and that $N$ is the order $ertmathcal{R}ert$ of the rotation group $mathcal{R}$ of one of the five Platonic polyhedra. We study motions that, up to a relabeling of the $N$ particles, are invariant under $mathcal{R}$. By variational techniques we prove the existence of periodic and chaotic motions

On the possible values of the orbit distance between a near-Earth asteroid and the Earth

We consider all the possible trajectories of a near-Earth asteroid (NEA), corresponding to the whole set of heliocentric orbital elements with perihelion distance q ≤ 1.3 au and eccentricity e ≤ 1 (NEA class). For these hypothetical trajectories, we study the range of the values of the distance from the trajectory of the Earth (assumed on a circular orbit) as a function of selected orbital elements of the asteroid. The results of this geometric approach are useful to explain some aspects of the orbital distribution of the known NEAs. We also show that the maximal orbit distance between an object in the NEA class and the Earth is attained by a parabolic orbit, with apsidal line orthogonal to the ecliptic plane. It turns out that the threshold value of q for the NEA class (qmax = 1.3 au) is very close to a critical value, below which the above result is not valid

On the existence of connecting orbits for critical values of the energy

We consider an open connected set Ω and a smooth potential U which is positive in Ω and vanishes on â\u88\u82Ω. We study the existence of orbits of the mechanical system u¨=Ux(u), that connect different components of â\u88\u82Ω and lie on the zero level of the energy. We allow that â\u88\u82Ω contains a finite number of critical points of U. The case of symmetric potential is also considered

Orbit determination with the two-body integrals. III

We present the results of our investigation on the use of the two-body integrals to compute preliminary orbits by linking too short arcs of observations of celestial bodies. This work introduces a significant improvement with respect to the previous papers on the same subject: citet{gdm10,gfd11}. Here we find a univariate polynomial equation of degree 9 in the radial distance $ho$ of the orbit at the mean epoch of one of the two arcs. This is obtained by a combination of the algebraic integrals of the two-body problem. Moreover, the elimination step, which in (Gronchi et al. 2010, 2011) was done by resultant theory coupled with the discrete Fourier transform, is here obtained by elementary calculations. We also show some numerical tests to illustrate the performance of the new algorithm