On the Singularities of Generalized Solutions to n-Body-Type Problems

Abstract

The validity of Sundman-type asymptotic estimates for collision solutions is established for a wide class of dynamical systems with singular forces, including the classical N -body problems with Newtonian, quasi-homogeneous and logarithmic potentials. The solutions are meant in the generalized sense of Morse (locally -in space and time-minimal trajectories with respect to compactly supported variations) and their uniform limits. The analysis includes the extension of the Von Zeipel's Theorem and the proof of isolatedness of collisions. Furthermore, such asymptotic analysis is applied to prove the absence of collisions for locally minimal trajectories.

Avoiding collisions under topological constraints in variational problems coming from celestial mechanics

“…This idea has been widely generalized in [10,11] in order to include symmetries, several potentials, and a large numer of applications to the search of periodic solutions to the N -body problem. Very general results on the absence of collisions for minimizers of the Bolza problem are reported in [2].…”

Nearly Parallel Vortex Filaments in the 3D Ginzburg–Landau Equations

“…Briefly, the point is that κ n (Ω) is constructed from the Bethuel-Brezis-Hélein renormalized energy on ω with natural boundary conditions, whereas κ n (Ω; g) uses the renormalized energy on ω with Dirichlet data u = g on ∂ω. The only places where this requires any changes in the proof of Theorem 3 are the following: 3 In the special case when ω is a ball, results similar to those of part (b) above are proved in [10].…”

“…As it is usual in this kind of construction, we define a sequence of trial maps (u ε ) based on canonical harmonic maps with prescribed singularities in ω, introduced in [4], section I. 3. This provides us with the right estimates for e 2d ε .…”

“…The system (1.17) appears in various contexts; its solutions are equilibria of reduced systems in fluid mechanics [23,22,26] and supplemented with suitable periodic conditions they correspond to trajectories in the planar n-body problem with logarithmic potential (examples of periodic orbits with or without collisions and for different potentials may be found in [8,15,7,3]). Theorem 2.…”

Platonic Polyhedra, Periodic Orbits and Chaotic Motions in the $$N$$ N -body Problem with Non-Newtonian Forces

“…The possibility that, when α < 2, a minimizer of the action presents collisions is one of the main obstructions to the variational approach to the existence of periodic motions for the N -body problem. We refer to [15,16] and [9] for various results on the problem of collisions.…”