Metric Thermodynamic Phase Space and Stability Problems

In this note, we study thermodynamic systems described in the Thermodynamic Phase Space (TPS), commonly referred as the contact geometry approach to thermodynamics. Following classical and recent contributions in the field, we try to fill a gap between the contact geometry endowed with a metric and stability problems considered in the literature. This examination leads to new interpretations of previously obtained results, and highlights new problems and avenue for research.