  A rigorous proof of a theorem on the coexistence of smooth Lyapunov function
and smooth planar dynamical system with one arbitrary limit cycle is given,
combining with a novel decomposition of the dynamical system from the
perspective of mechanics. We base on this dynamic structure incorporating
several efforts of this dynamic structure on fixed points, limit cycles and
chaos, as well as on relevant known results, such as Schoenflies theorem,
Riemann mapping theorem, boundary correspondence theorem and differential
geometry theory, to prove this coexistence. We divide our procedure into three
steps. We first introduce a new definition of Lyapunov function for these three
types of attractors. Next, we prove a lemma that arbitrary simple closed curve
in plane is diffeomorphic to the unit circle. Then, the strict construction of
smooth Lyapunov function of the system with circle as limit cycle is given by
the definition of a potential function. And then, a theorem is hence obtained:
The smooth Lyapunov function always exists for the smooth planar dynamical
system with one arbitrary limit cycle. Finally, by discussing the two criteria
for system dissipation(divergence and dissipation power), we find they are not
equal, and explain the meaning of dissipation in an infinitely repeated motion
of limit cycle.
