We present a comprehensive analysis of the dynamic and thermodynamic stability of neutron stars composed of superconducting-superfluid mixtures within the Iyer-Wald formalism. We derive the first law of thermodynamics and the necessary and sufficient condition under which dynamic equilibrium implies thermodynamic equilibrium. By constructing the phase space and canonical energy, we show that the dynamic stability for perturbations, restricted in symplectic complement of trivial perturbations with the Arnowitt-Deser-Misner 3-momentum unchanged, is equivalent to the non-negativity of the canonical energy. Furthermore, dynamic stability against restricted axisymmetric perturbations guarantees the dynamic stability against all axisymmetric perturbations. We also prove that the positivity of canonical energy on all axisymmetric perturbations within the Lagrangian displacement framework with fixed angular momentum is necessary for thermodynamic stability. In particular, the equivalence of dynamic and thermodynamic stability is established for radial perturbations of static, spherically symmetric isentropic configurations.

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• Neutron stars & pulsars