"For what classes of ODEs, describing dynamical systems, does the Lyapunov’s Second Method formulated in the classical and canonically generalized forms define the necessary and sufficient conditions for the (asymptotical) stability of motion?" -- Wikipedia.org.

"Liapunov's Second Method is a generalization to the theory of nonlinear systems of ODEs of two basic physical principles: A state of a conservative physical system is stable only if its potential energy has a local minimum at that state."

Reference Links:

https://math.okstate.edu/people/binegar/4233/4233-l06.pdf;

https://www.researchgate.net/project/Important-Question-about-the-Lyapunovs-Second-Method-for-Stability.