PIMS Marsden Memorial Lecture: An Octahedral Gem Hidden in Newton's Three Body Problem
- Date: 07/25/2012
Richard Montgomery (UC Santa Cruz)
An Octahedral Gem Hidden in Newton’s Three Body Problem
The three-body problem of Newton has singularities and symmetries. Singularities arise from collisions. McGehee  showed how to eliminate the triple collision singularity through a blow-up procedure. Levi-Civita  showed how to eliminate the binary collision singularities through a regularization procedure. Lagrange , followed by a host of others, showed how to eliminate symmetries through a reduction procedure. We sketch how a systematic and democratic (no body selected as being ‘special’) application of all three procedures leads to a regularized reduced three body problem which consists of a complete but non-Hamiltonian vector field on a manifold with boundary. Upon composing the transformations associated to reduction and regularization we discover a certain degree 4 map of the two-sphere which is related to the octahedron. The vertices of the octahedron represent binary collisions. When all masses are equal, this octahedron leads to a discrete symmetry group of order 48 for the regularized reduced vector field. Does a global understanding of this completed vector field, the manifold on which it lives, or of the role of the octahedron, help us to answer any of the long- outstanding basic open questions regarding the three body problem? How much of this story generalizes to N bodies moving in d dimensions? (report on joint work with Rick Moeckel)
Richard Montgomery received undergraduate degrees in
both mathematics and physics from Sonoma State in Northern California.
He completed his PhD under Jerry Marsden at Berkeley in 1986, after
which he held a Moore Instructorship at MIT for two years, followed by
two years of postdoctoral studies at University of California, Berkeley.
Montgomery's research fields are geometric mechanics, celestial
mechanics, control theory and differential geometry and is perhaps best
known for his rediscovery - with Alain Chenciner - of Cris Moore's
figure eight solution to the three-body problem, which led to numerous
new 'choreography' solutions. He also established the existence of the
first-known abnormal minimizer in sub-Riemannian geometry, and is known
for investigations using gauge-theoretic ideas of how a falling cat
lands on its feet. He has written one book on sub-Riemannian geometry.
The PIMS Marsden Memorial Lecture Series is dedicated to the memory of
Jerrold E Marsden (1942-2010), a world-renowned Canadian applied
mathematician. Marsden was the Carl F Braun Professor of Control and
Dynamical Systems at Caltech, and prior to that he was at the University
of California, Berkeley, for many years. He did extensive research in
the areas of geometric mechanics, dynamical systems and control theory.
He was one of the original founders in the early 1970s of reduction
theory for mechanical systems with symmetry, which remains an active and
much studied area of research today.
The inaugural Marsden Memorial Lecture was given by Alan Weinstein (University of California, Berkeley) in July of 2011 at ICIAM in Vancouver.
Mark J Gotay (PIMS / UBC)
This lecture will take place during the conference
on "Geometry, Symmetry, Dynamics, and Control: The Legacy of Jerry
Marsden" at the Fields Institute.
View the lecture online at mathtube.org