The classical Poincaré Normal Form Theorem asserts that a singular point of an analytic planar vector field is a non-degenerate center if and only if, after an analytic change of coordinates, the system can be written in the rotational normal form \( f(x^{2}+y^{2})\bigl (y\,\partial _{x}-x\,\partial _{y}\bigr ), \qquad f(0)>0. \) In this paper we prove that every analytic planar vector field with a non-degenerate center at the origin is locally analytically conjugate to a one-degree-of-freedom mechanical Hamiltonian system \( y\,\partial _{x}-V'(x)\,\partial _{y}, \) where V is analytic and satisfies \(V(0)=V'(0)=0\) and \(V''(0)>0\) . The construction of V is completely explicit and depends solely on the period function of the original center. Consequently, the local analytic classification of non-degenerate centers reduces to the classification of analytic potentials, or equivalently, of their period functions. Our result provides a local analytic answer to a question related to Chicone’s 1987 work, where he established a celebrated criterion for studying the monotonicity of the period function of mechanical Hamiltonian systems using only the potential V and its derivatives \(V'\) , \(V''\) , and \(V'''\) . In this sense, our theorem shows that the local monotonicity problem for the period function of an arbitrary analytic vector field with a non-degenerate center reduces to the monotonicity problem for the period function of an associated mechanical system.
