Quadratic forms diagonalization methods can be used in addressing the stability of physical systems. Thermodynamic stability conditions appears as an eigenvalues fundamental problem, in particular when postulational approaches is taken. The second-order derivatives or appropriate relations between such derivatives of the energy, entropy or any considered thermodynamic potential, as Helmholtz, enthalpy and Gibbs, have interesting mathematical features that directly imply in the physical stability, obtained by use and as consequence of analytical techniques. Formal aspects on the thermal and mechanical stability become simple consequences, but no less formal, of the superposition of rigorously established physical laws, and appropriate applications of mathematical techniques.
Part of the book: Recent Developments in the Solution of Nonlinear Differential Equations