Abstract 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) (y\, ₗ-x\, ₘ), f (0) >0. f (x 2 + y 2) (y ∂ x - x ∂ y), 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\, ₗ-V' (x) \, ₘ, y ∂ x - V ′ (x) ∂ y, where V is analytic and satisfies V (0) =V' (0) =0 V (0) = V ′ (0) = 0 and V'' (0) >0 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 ′, V'' V ′ ′, and V''' 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.
Francisco J. S. Nascimento (Sat,) studied this question.