In Arnold's Classical Mechanics of Classical Mechanics, he refers to Lyapunov stability in many of the problems in the second chapter.
E.g. on page 20: "Problem: Consider a periodic motion along the closed phase curve corresponding to the energy level $E$. Is it stable in the sense of Lyapunov? Answer: No."
On page 36: "Problem: For which values of $\alpha$ is motion along a circular orbit in the field with potential energy $U = r^\alpha$, $-2\leq\alpha\leq\infty$, Lyapunov stable?"
In the footnote he refers to his own book on Ordninary Differential Equations for a definition of Lyapunov stability. In that book he defines Lyapunov stability ONLY for equilibrium points of the equation, i.e. if
$$ \dot{\mathbf{x}} = \mathbf{v} \left( \mathbf{x} \right) $$ where $\mathbf{v}$ is an at least twice differentiable vector field. Then we can only talk about Lyapunov stability if we are looking at a solution, i.e. a solution where the vector field is zero.
(I.e. if $\mathbf{v}(\mathbf x_0) = \mathbf 0$ we obviously will get a solution $\mathbf x(t) = \mathbf x_0$ if the initial condition is $\mathbf x(0) = \mathbf x_0$.)
My confusion is that none of the two systems I in the problems above seem to be in equilibrium positions! For example, the motion in a central potential is described by
$$ \left( \begin{array}{c} \dot{\mathbf r} \\ \dot{\mathbf v} \end{array} \right) = \left( \begin{array}{c} \mathbf v \\ - \Phi (r) \mathbf{e}_r \end{array} \right) $$ So wouldn't an equilibrium point be one for which the right hand side of these equations are zero?
Using polar coordinates I cannot even manage to write the equations for $r$ and $\phi$ in the form above...
As you can tell from my rambling I am clearly confused about all this and I am most likely missing something essential here.