A concise exposition of the results to be expounded in this memoir will be
offered in the Introduction. While some of these results are sketched here, the
main task at hand is to describe, informally and in plain language, the questions
and problems motivating our work, as well as the ideas and point of view leading
us ultimately to answers. It will be illustrative to keep two examples in mind that
serve as good prototypes for the larger class of systems that will be studied later.
T w o simple Hamiltonia n s y s t e m s
E x a m p l e A . Consider the motion of a point mass m moving through ordi-
nary three-dimensional space, free of external forces including gravity. The linear
momentum Jijn of the mass is conserved, so that the motion is in a straight line at
Exampl e B . Alternatively, suppose that our point mass is constrained to move
on the surface of a smooth sphere, the only external force being the normal force
necessary to maintain the constraint. In this case the angular momentum J
a n g
conserved and the motion is along great circles of the sphere at constant speed.
Both Examples A and B possess a conservation law. The conservation laws can
be 'explained' by the existence of symmetries in the underlying equations. This is
the content of a well-known theorem of Noether (see, e.g., Marsden (1992)). Exam-
ple A clearly possesses a three-dimensional translational symmetry, while Example
B possesses a rotational symmetry.
Suppose these symmetries are broken. For instance, suppose that in Example
A we add a weak spatially periodic potential, and in Example B we introduce a
small aspherical distortion. What is the fate of the above conservation laws?