  We provide a constructive proof on the equivalence of two fundamental
concepts: the global Lyapunov function in engineering and the potential
function in physics, establishing a bridge between these distinct fields. This
result suggests new approaches on the significant unsolved problem namely to
construct Lyapunov functions for general nonlinear systems through the analogy
with existing methods on potential functions. In addition, we show another
connection that the Lyapunov equation is a reduced form of the generalized
Einstein relation for linear systems.
