In the Energy-Casimir method, it is said we look for 'conserved quantities' that are a function of the system. Then subsequently $V = \mathcal{H} + \mathcal{C}$ becomes the candidate Lyapunov function, in which $\mathcal{H}$ is the Hamiltonian and $\mathcal{C}$ is the Casimir function.
But how does it make sense that you see $\dot{\mathcal{C}}=0$ in books? It seems that if $\mathcal{C}$ remains constant that this will not move the coordinates of the minimum of $V$ at all. But moving it is what you want to get stability (proof) at a certain (equilibrium) point of the system.
Can anybody explain this?