The collinear problem forms a planar slice of the full planar three-body problem. Figure courtesy of Rick Moeckel.
We will discuss several specific projects at the interface of mechanics, geometry and analysis.
The fascinating phenomenon of stabilization by vibration suggests one group of problems.
The most famous example of such a stabilization is the Kapitsa pendulum in which the upside-down
unstable equilibrium of the standard pendulum becomes a stable equilibrium when the pendulum's pivot is
vibrated vertically at a high enough frequency. See, for instance, the following YouTube video:
This effect led to the invention of the cyclotron and of the Paul trap, for which W. Paul received the 1989 Nobel Prize in physics.
Another effect in the same spirit is the stabilization of a viscous fluid by vibration.
A surface of molasses in an appropriately vibrating container can be made to form a vertical wall!
One proposed activity will be to recast these problems in terms of differential geometry, as the study of geodesics on
vibrating manifolds. This recasting has not been widely explored despite its important applications.
In celestial mechanics vibrational stablization questions also arise.
For example, the equal mass planar three-body problem (a three degree of freedom system after reductions) contains
four invariant submanifolds of codimension 2: the collinear three-body problem and its three isosceles sub-problems.
These four sub-problems are much better understood than the full planar problem. Oscillations orthogonal to their
submanifolds have the potential of connecting the submanifolds in interesting and not well-understood ways.
Or perhaps such connections are blocked in some way. From the perspective of differential geometry, we have a
Riemannian three-manifold which contains 4 distinguished totally geodesic surfaces.
What can we say about the growth or oscillation of the normal mode (orthogonal to the surface)
of the Jacobi equation for geodesics lying in one of these surfaces? How does this understanding of
normal modes lead to a better understanding of the full geodesic flow?
The discussion will greatly benefit from collaboration of people with diverse interests ranging from
geometry to differential equations to mechanics.