TY - JOUR

T1 - An example exempted from Thomson-Tait-Chetayev's theorem

AU - Paerhati, Abuduwaili

AU - Fukumoto, Yasuhide

N1 - Copyright:
Copyright 2013 Elsevier B.V., All rights reserved.

PY - 2013/4

Y1 - 2013/4

N2 - An example is given of a mechanical system whose behavior does not follow Thomson-Tait-Chetayev's (TTC) theorem which states that, for a system with an unstable potential, a state stabilized by gyroscopic forces goes unstable after the addition of arbitrary dissipation. The example is brought by a system, closely related with a heavy symmetrical top, describing motion of a charged spherical pendulum subjected to the Lorentz force, in a magneticmonopole field sitting at the sphere center, as well as the gravity force. A drag force proportional to the velocity is exerted on the pendulum. The upright state, an equilibrium stabilized by the Lorentz force, is shown to be exempted from the TTC theorem. The numerical calculation of the full nonlinear system is performed for precession. For the slow precession, the drag force acts to continuously tilt down the top axis toward vertically downward equilibrium, following the dissipation-induced instability. On the contrary, for the fast precession, the drag acts to continuously tilt up the axis against the gravity force, despite losing the energy.

AB - An example is given of a mechanical system whose behavior does not follow Thomson-Tait-Chetayev's (TTC) theorem which states that, for a system with an unstable potential, a state stabilized by gyroscopic forces goes unstable after the addition of arbitrary dissipation. The example is brought by a system, closely related with a heavy symmetrical top, describing motion of a charged spherical pendulum subjected to the Lorentz force, in a magneticmonopole field sitting at the sphere center, as well as the gravity force. A drag force proportional to the velocity is exerted on the pendulum. The upright state, an equilibrium stabilized by the Lorentz force, is shown to be exempted from the TTC theorem. The numerical calculation of the full nonlinear system is performed for precession. For the slow precession, the drag force acts to continuously tilt down the top axis toward vertically downward equilibrium, following the dissipation-induced instability. On the contrary, for the fast precession, the drag acts to continuously tilt up the axis against the gravity force, despite losing the energy.

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U2 - 10.7566/JPSJ.82.043002

DO - 10.7566/JPSJ.82.043002

M3 - Article

AN - SCOPUS:84876004594

VL - 82

JO - Journal of the Physical Society of Japan

JF - Journal of the Physical Society of Japan

SN - 0031-9015

IS - 4

M1 - 043002

ER -