Some problems on the classical n-body problem

On the Existence of Collisionless Equivariant Minimizers for the Classical n-body Problem

We show that the minimization of the Lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n-body problem, provided that some simple conditions on the symmetry group are satisfied. More precisely, we give a fairly general condition on symmetry groups G of the loop space Λ for the n-body problem (with potential of homogeneous degree −α, w...

The N-body Problem

New Perspectives on the N -body Problem

The problem of the motion of N ≥ 2 point–masses (i.e., ideal bodies with no physical dimension identified with points in the Euclidean three–dimensional space) interacting only through Newton’s law of mutual gravitational attraction, has been a central issue in astronomy, physics and mathematics since the early developments of modern calculus. When N = 2 the problem has been completely solved (...

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 Dl, where Dl is the group of order 2l generated by two rotations of angle π around two secant lines in space meeting at an angle of π/l. By adding a homogeneous gravitational (Newtonian) potential one finds a special n-body proble...

The quantum N-body problem

This selective review is written as an introduction to the mathematical theory of the Schro ̈dinger equation for N particles. Characteristic for these systems are the cluster properties of the potential in configuration space, which are expressed in a simple geometric language. The methods developed over the last 40 years to deal with this primary aspect are described by giving full proofs of...