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This talk will discuss the application
of dynamical systems and mechanics to the classical three body problem,
the resonant transition and temporary capture of comets and to the design
of space missions such as the {\it Genesis Discovery Mission}: http://genesismission.jpl.nasa.gov/
An exciting recent development is the observation that many comets and
upcoming space missions make use of structures in the dynamics of the
three body problem which dynamical systems researchers have been investigating
for their intrinsic interest. These structures involve heteroclinic and
homoclinic connections between periodic orbits in the problem.
We shall describe a numerical proof of the existence of a heteroclinic
connection between pairs of periodic orbits, one around the libration
point $L_1$ and the other around $L_2$, with the two periodic orbits having
the same energy. This result is applied to the resonance transition problem
and to the explicit numerical construction of interesting orbits with
prescribed itineraries.
The point of view developed is that the invariant manifold structures
associated to $L_1$ and $L_2$ as well as the aforementioned heteroclinic
connection are fundamental tools that can aid in understanding dynamical
channels throughout the solar system as well as transport between the
``interior'' and ``exterior'' Hill's regions and other resonant phenomena.
Using these tools, a new technique for constructing missions, such as
a petit grand tour of the moons of Jupiter will be given. Other issues
such as the use of variational integration algorithms and optimal control
techniques for low thrust missions will also be discussed.
We will also briefly describe why it is that these same techniques are
also relevant to the study of transition states for the ionization of
a hydrogen atom interacting with combinations of external electric and
magnetic fields.
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