Mechanical Normal Forms for Analytic Centers

• Autores: Francisco J.S. Nascimento
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 2, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • 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 (x2 + y2) 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 ∂x − V (x) ∂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.

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