  The stability of static solutions of the spherically symmetric,
asymptotically flat Einstein-Vlasov system is studied using a Hamiltonian
approach based on energy-Casimir functionals. The main result is a coercivity
estimate for the quadratic part of the expansion of the natural energy-Casimir
functional about an isotropic steady state. The estimate shows in a quantified
way that this quadratic part is positive definite on a class of linearly
dynamically accessible perturbations, provided the particle distribution of the
steady state is a strictly decreasing function of the particle energy and
provided the steady state is not too relativistic. This should be an essential
step in a fully non-linear stability analysis for the Einstein-Vlasov system.
In the present paper it is exploited for obtaining a linearized stability
result.
