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In the presence of rf focusing and a purely inductive impedance bunch, equilibria in the form of Haïssinski distributions—when they exist—are linearly stable. This is the case whether the potential well distortion associated with the impedance causes bunch lengthening or shortening. We provide a general proof of this fact using Hamiltonian methods and energy principles. In the presence of bunch shortening our analysis indicates that there is a critical current for linear stability. However, this threshold is identical to the critical current defining the condition for the very existence of a Haïssinski equilibrium

Stability analysis of longitudinal beam dynamics using noncanonical Hamiltonian methods and energy principles