  The Helmholtz-Hodge decomposition (HHD) is applied to the construction of
Lyapunov functions. It is shown that if a stability condition is satisfied,
such a decomposition can be chosen so that its potential function is a Lyapunov
function. In connection with the Lyapunov function, vector fields with strictly
orthogonal HHD are analyzed. It is shown that they are a generalization of
gradient vector fields and have similar properties. Finally, to examine the
limitations of the proposed method, planar vector fields are analyzed.
