Navegando por Assunto "Sistemas potenciais"

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    Conjugação analítica entre sistemas diferenciais planares e sistemas potenciais
    (2024-12-19) Nacimento, Francisco José dos Santos; https://orcid.org/0009-0004-9536-7979; http://lattes.cnpq.br/9091581147583991
    The classic Poincaré Normal Form Theorem states that a critical point of an analytic planar vector field is a non-degenerate center if and only if there is an analytic coordinate change such that in the new coordinates the vector field initial is of the form $f(x^2+y^2)\big(y\frac{\partial}{\partial x}-x\frac{\partial}{\partial y}\big)$, where $f$ is an analytic function defined in a neighborhood of the origin such that $f(0)>0.$ In this article it is proved that an analytical planar vector field with a non-degenerate center at $(0,0)$ is analytically conjugate, in a neighborhood of $(0,0)$, to a Hamiltonian vector field of the form $y\frac{\partial}{\partial x}-V'(x)\frac{\partial}{\partial y}$, where $V$ is an analytic function defined in a neighborhood of the origin such that $V(0)=V'(0)=0$ and $V''(0)>0.$ This result is a partial answer to a problem proposed by Chicone in 1987.