A theorem in symplectic geometry
No. 241, 1986
(Symplectic geometry)
India ink on paper, 22x31cm.
In a space of spheres, soap bubbles of all sizes seek to fill every gap, creating a region where no space remains, pressing together, touching on at most one point. Now consider the proof of the following theorem: On an arbitrary smooth symplectic manifold of dimension 2n, there always exists a collection of n smooth independent functions in involution. Such families of functions are called complete involutive families. In Hamiltonian mechanics, where such functions permit one to integrate Hamiltonian differential equations, the search for such families is a central problem. To construct such a family, one must cover nearly the whole manifold M 2" with disjoint open balls so that the complement of the union of these balls has no volume. This picture shows a countable family of balls of just such a kind, where the spheres get progressively smaller, filling all free space with ever finer orbs. From this pattern, we construct a complete involutive family of functions on one of the balls, duplicating the process on all remaining balls. On each ball then we get the necessary family of functions, leaving only the task of sewing them together, with each function vanishing on the ball's boundary.
