Brouwer's problem on a heavy particle in a rotating vessel: wave propagation, ion traps, and rotor dynamics
arXiv:1012.0892 · doi:10.1016/j.physleta.2011.02.056
Abstract
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 singularity corresponding to the quadruple zero eigenvalue. In the presence of dissipative and non-conservative positional forces there is a couple of Whitney umbrellas 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 dissipative and non-conservative positional forces.
18 pages, 6 figures
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Cited by in corpus (5)
- Rotating saddle trap as Foucault's pendulum
- Rotating saddle trap: A Coriolis force in an inertial frame
- Geometry of escape and transition dynamics in the presence of dissipative and gyroscopic forces in two degree of freedom systems
- PT-symmetry, indefinite damping and dissipation-induced instabilities
- From rotating fluid masses and Ziegler's paradox to Pontryagin- and Krein spaces and bifurcation theory