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We provide a new derivation of the conditions of dynamical andthermodynamical stability of homogeneous and inhomogeneous isothermaldistributions in the Hamiltonian Mean Field (HMF) model. This proof completesthe original thermodynamical approach of Inagaki [Prog. Theor. Phys. 90, 557(1993)]. Our formalism, based on variational principles, is simple and themethod can be applied to more general situations. For example, it can be usedto settle the dynamical stability of polytropic distributions with respect tothe Vlasov equation [Chavanis & Campa, arXiv:1001.2109]. For isothermaldistributions, the calculations can be performed fully analytically, providingtherefore a clear illustration of the method.

Dynamical and thermodynamical stability of isothermal distributions in the HMF model