7 research outputs found
On the stability problem for the free rigid body
In the general case of the free rigid body we give a list
of integrals of motion, which generate the set of Mishchenko's integrals. In
the case of we prove that there are fifteen coordinate type
Cartan subalgebras which on a regular adjoint orbit give fifteen Weyl group
orbits of equilibria. These coordinate type Cartan subalgebras are the
analogues of the three axes of equilibria for the classical rigid body on
. The nonlinear stability and instability of these equilibria
is analyzed. In addition to these equilibria there are ten other continuous
families of equilibria.Comment: Preprint of an article submitted for consideration in International
Journal of Geometric Methods in Modern Physics \copyright 2011 copyright
World Scientific Publishing Company http://www.worldscinet.com/ijgmmp
Instability conditions for circulatory and gyroscopic conservative systems
We give a method which generates sufficient conditions for instability of
equilibria for circulatory and gyroscopic conservative systems. The method is
based on the Gramians of a set of vectors whose coordinates are powers of the
roots of the characteristic polynomial for the studied systems. New instability
results are obtained for general circulatory and gyroscopic conservative
systems. We also apply this method for studying the instability of motion for a
charged particle in a stationary electromagnetic field