Colloquium
3:30 p.m., Friday
Math 100
Richard G. Froese
Department of Mathematics
UBC
Realizing holonomic constraints in classical and quantum mechanics
A motion of a particle constrained to lie on a submanifold of
configuration space is governed by wellknown differential equations.
For example, if the particle is allowed to move freely on the
submanifold, the classical orbits are geodesics, while the quantum
motion is governed by the LaplaceBeltrami operator.
One may try to model such a constraint by allowing the particle to
move in the whole configuration space, subject to a very large
force that pushes the particle onto the submanifold. As this
constraining force becomes larger and larger, do the orbits
converge to solutions of the wellknown equations on the submanifold?
Surprisingly, the answer is often no. Sometimes the particles obey
an equation with extra potential terms, and sometimes the observed
motion doesn't seem to obey an equation at all. Understanding the
limiting motion involves quantum and classical versions of averaging,
as well as the geometry of the normal bundle.
