# Poincare's last geometric theorem

Which problem in celestial mechanics led Poincare to his conjecture about fixed points of area preserving maps of the annulus to itself? I believe he was working on some differential equations. What was the problem and what was the differential equations he was working on?

Poincare was not considering any differential equations, but instead the Jacobi integral $C_J$, which here takes the form $$C_J=-\frac{2H}{m}=-\left(\dot{x}^2+\dot{y}^2\right)+\left(x^2+y^2\right)+2\left(\frac{1-\mu}{r_1}+\frac{\mu}{r_2}\right)$$ for Hamiltonian $H$, test mass $m$, coordinates $(x,y)$, masses of main bodies $1-\mu$ and $\mu$, and distances to those bodies $r_1$ and $r_2$.