Mechanical Normal Forms for Analytic Centers
| dc.citation.volume | 25 | |
| dc.contributor.author | Nacimento, Francisco José dos Santos | |
| dc.contributor.authorID | https://orcid.org/0009-0004-9536-7979 | |
| dc.contributor.authorLattes | http://lattes.cnpq.br/9091581147583991 | |
| dc.date.accessioned | 2026-07-27T17:59:13Z | |
| dc.date.available | 2026-07-27T17:59:13Z | |
| dc.date.issued | 2026-02-28 | |
| dc.description.resumo | 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. | |
| dc.identifier.doi | https://link.springer.com/article/10.1007/s12346-026-01469-1 | |
| dc.identifier.uri | https://repositorio.univasf.edu.br/handle/123456789/1493 | |
| dc.relation.titlePeriodicals | Qualitative Theory of Dynamical Systems | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.subject | Non-degenerate center · Period function · Analytic conjugacy · Mechanical Hamiltonian system | |
| dc.subject.cnpq | CIENCIAS EXATAS E DA TERRA | |
| dc.title | Mechanical Normal Forms for Analytic Centers | |
| dc.type | Artigo de Periódico |