  The main goal of this paper is to prove that if the energy-momentum (or
energy-Casimir) method predicts formal instability of a relative equilibrium in
a Hamiltonian system with symmetry, then with the addition of dissipation, the
relative equilibrium becomes spectrally and hence linearly and nonlinearly
unstable. The energy-momentum method assumes that one is in the context of a
mechanical system with a given symmetry group. Our result assumes that the
dissipation chosen does not destroy the conservation law associated with the
given symmetry group---thus, we consider internal dissipation. This also
includes the special case of systems with no symmetry and ordinary equilibria.
The theorem is proved by combining the techniques of Chetaev, who proved
instability theorems using a special Chetaev-Lyapunov function, with those of
Hahn, which enable one to strengthen the Chetaev results from Lyapunov
instability to spectral instability. The main achievement is to strengthen
Chetaev's methods to the context of the block diagonalization version of the
energy momentum method given by Lewis, Marsden, Posbergh, and Simo. However, we
also give the eigenvalue movement formulae of Krein, MacKay and others both in
general and adapted to the context of the normal form of the linearized
equations given by the block diagonal form, as provided by the energy-momentum
method. A number of specific examples, such as the rigid body with internal
rotors, are provided to illustrate the results.
