Mechanical Normal Forms for Analytic Centers
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2026-02-28
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Campus
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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.