3
$\begingroup$

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?

$\endgroup$
3
$\begingroup$

Poincare was studying the circular restricted three-body problem, where two bodies in a binary system revolve around their barycenter and a third test mass moves in some orbit around them. Birkhoff acknowledges this in his proof of the theorem, Proof of Poincare's Geometric Theorem. Specifically, Poincare was looking to prove that this third body may have periodic orbits.

John Franks is quoted in Steve Nadis' A History in Sum as describing the annulus as "a two-dimensional slice of three-dimensional space". If a planet moves from a point on the annulus to the rest of the space and then back, this must be a fixed point, and there must therefore be periodicity. In fact, Poincare's theorem states that if there is one such point there must be a second point; one of the two describes the third body.

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 $r_2$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.