In this Brief Report we give the proof that the solution of any static test charge distribution in Schwarzschild space is unique. In order to give the proof we derive the first Green's identity written with p-forms on (pseudo) Riemannian manifolds. Moreover, the proof of uniqueness can be shown for either any purely electric or purely magnetic field configuration. The spacetime geometry is not crucial for the proof.

Weber's transformation is used to show how Lin's constraint should be replaced if fluid equations are derived from Hamilton's principle. The same technique is used to derive a three-circulation theorem and a generalization of Ertel's theorem for perfect multifluid plasmas. The Hamiltonian and Lagrangian formulation of the equations for fluid and electromagnetic potentials is given, with a discussion of their multivaluedness and their gauge and time dependence for static magnetohydrodynamic equilibria. The linear stability of these equilibria is shown to depend on the weight of a single negative eigenvalue of the internal energy variation, compared with all other (positive) contributions to the "energy" functional.

Vlasov’s equation and the ideal multifluid equations are considered in manifestly covariant form. In the latter case, a thermodynamic closure (locally the first law of thermodynamics) leads to a generalized Kelvin/Helmholtz theorem. In the former case, the local dispersion relation for Langmuir waves in a strong gravitational field is derived and solved.

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