Bounded orbits for 3 bodies in $\mathbb{R}^4$

TL;DR

This paper proves that in the 3-body problem in four-dimensional space, the energy function attains its minimum on bounded orbits, extending previous work on stable relative periodic orbits.

Contribution

It establishes the existence of minimal energy configurations for the 3-body problem in four dimensions, showing the energy cannot be minimized on unbounded sequences.

Findings

Energy cannot tend to its infimum on unbounded sequences

The infimum of energy is actually attained as a minimum

Supports existence of stable relative periodic orbits in 4D

Abstract

We consider the Newtonian 3-body problem in dimension 4, and fix a value of the angular momentum which is compatible with this dimension. We show that the energy function cannot tend to its infimum on an unbounded sequence of states. Consequently the infimum of the energy is its minimum. This completes our previous work \cite{AD19} on the existence of Lyapunov stable relative periodic orbits in the 3-body problem in $\mathbb{R}^4$.