  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.
