View source for Stability of Hamiltonian equilibria

====Example: the Stark problem====
An example of a relative equilibrium is the classical Stark problem
(Howard, 1995a),
in which a hydrogen atom is perturbed by a uniform electric field <math>{\mathbf F} = F \hat e_{z}</math> in the symmetry axis direction.
The Hamiltonian becomes \eqref{ReducedHam}  with effective potential
:<math>
      U^e(\rho,z) = \frac{\mu} {2\rho^2} - \frac{1}{r} - z,
</math>
<math> r = \sqrt{\rho^2 + z^2}</math>, <math>\mu</math> the scaled value of the conserved <math>\phi</math>-component of angular momentum, and <math>m = F = 1</math> in scaled atomic units.

This integrable system has ''relative equilibria'' at zero momentum and any
critical point of the effective potential,  <math>\nabla U^e(\rho_0,z_0) = 0 </math>, giving
:<math> 
      \rho_0^4 = \mu r_0^3,~~z_0 = r_0^3 \implies r_0(1-r_0^4)^2 = \mu .
</math>
Since the kinetic energy is positive definite, stability is completely determined
from the Hessian determinant, <math>\Delta = \det D^2 U^e</math>: it is not necessary to calculate the eigenvalues
(this is implicit in the analysis of the Hessian).
<figref>SPAFig4.jpg</figref> shows level sets of <math> U^e </math> in
scaled coordinates for three values of the angular momentum.  Stable circular orbits occur when <math>\Delta_0 > 0</math>, corresponding to a local minimum of <math>U^e</math>  if <math> U_{\rho\rho}^e >0 </math> and a local maximum if <math> U_{\rho\rho}^e <0</math>. A saddle point, where <math> \Delta_0 < 0</math>,
corresponds to an unstable circular orbit.
Since <math> H </math> is a natural two degree-of-freedom system, Krein bifurcations are impossible and stability can only be lost
via a saddle-node or pitchfork bifurcation.
[[Image:SPAFig4.jpg|thumb|700px|left|stab2D|Level sets of <math> U^e </math> for the Stark Problem (a) <math>\mu<\mu^* \ ,</math> (b) <math> \mu = \mu^* \ ,</math> (c) <math> \mu > \mu^* \ .</math>]]

A critical point changes type when <math> \Delta_0 </math> passes through zero.
It follows that <math> \Delta_0 </math> changes sign when <math> \mu = \mu^* = 64\sqrt{3}/243 \ ,</math> at which point a stable-unstable pair of equilibria merge and disappear for <math> \mu > \mu^*</math>, as seen
in <figref>SPAFig4.jpg</figref>. Since <math>H</math> is natural one obtains as an added bonus the Lyapunov stability of the elliptic points. 

Other physically interesting, axisymmetric systems possessing relative equilibria include ion motion in Paul traps (Bluemel, 1995), 
planetary dust dynamics (Howard et. al., 1999), the problem of two fixed centers, and galactic dynamics.