Potential wells with a unique brake orbit. Counterexamples to a conjecture by H. Seifert

TL;DR

This paper constructs counterexamples to Seifert's 1948 conjecture by demonstrating Hamiltonian systems with a unique brake orbit at certain energy levels, challenging the belief that multiple brake orbits must exist.

Contribution

It provides explicit examples of Hamiltonian systems with a single brake orbit at specific energy levels, disproving Seifert's conjecture.

Findings

Existence of systems with only one brake orbit at certain energies

Counterexamples to Seifert's conjecture from 1948

Potential implications for the study of Hamiltonian dynamics

Abstract

In this paper we prove the existence of real-analytic natural Hamiltonian systems - i.e. where H(q,p)=T(q,p)+V(q) in the 2N-dimensional real space, where N is any integer greater than 1 - with non critical energy levels E for the potential V such that the sublevel E of V is homeomorphic to the N-dimensional disk, and that only one brake orbit of energy E exists. A famous conjecture formulated by H. Seifert in 1948 claimed the existence of at least N distinct brake orbits for this situation.