Publications Repository - Helmholtz-Zentrum Dresden-Rossendorf

1 Publication
L.E.J. Brouwer's heavy particle in a rotating vessel and ion traps: a curious dissipative system with pure imaginary eigenvalues
Kirillov, O.;
In 1918 Brouwer considered stability of a heavy particle in a rotating vessel. This was the first demonstration of a rotating saddle trap which is a mechanical analogue for quadrupole particle traps of Penning and Paul. We revisit this pioneering work in order to uncover its intriguing connections with classical rotor dynamics and fluid dynamics, stability theory of Hamiltonian and non-conservative systems as well as with the modern works on crystal optics and atomic physics. In particular, we find that the boundary of the stability domain of the undamped Brouwer's problem possesses the Swallowtail-like singularity corresponding to the quadruple zero eigenvalue. In the presence of dissipative non-conservative forces there is a couple of exceptional points in the spectrum that correspond to the Whitney umbrella singularities on the boundary of the asymptotic stability domain. The handles of the umbrellas form a set where all eigenvalues of the system are pure imaginary despite the presence of the dissipative non-conservative forces. This classical dissipative system demonstrates a non-trivial connection between the regions with pure imaginary and complex spectrum in the space of parameters that may give a useful insight to how regions with real and complex spectrum could be connected in the case of near PT-symmetric Hamiltonians.
Keywords: Rotating saddle trap, dissipation, Lyapunov stability, gyroscopic stabilization, multiple eigenvalues, exceptional point, stability boundary
  • Invited lecture (Conferences)
    Quantum Physics with Non-Hermitian Operators, International Seminar and Workshop, 15.-25.06.2011, Dresden, Deutschland

Publ.-Id: 15811 - Permalink