On the Lyapunov instability in Lagrangian dynamics

The paper proves Lyapunov instability under the hypotheses dU(∇ρ f)=O(U) and ||ι_{∇ρ f} dµ||=O(U^{1/2}) via a time-rescaling and a gradient-flow–adapted coordinate argument that yields a positive drift of f along solutions and hence exit from any small neighborhood. The candidate solution establishes the same instability using an energy-based differential inequality for h(t)=df_x(v), integrates it twice, and shows exit from thin f-slabs for arbitrarily small initial velocities. The approaches are logically consistent and reach the same conclusion; the model’s derivation has a minor constant slip in bounding the magnetic term but is easily fixed without affecting correctness.