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{\Large \bf Symplectic Mappings}\\
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{\small John D. Hadjidemetriou}\\
{\small Department of Physics, University of Thessaloniki, Greece}
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{\bf Abstract.} A widely used method to study the evolution of a
Hamiltonian system of $n$ degrees of freedom in phase space is by the
method of Poincar\'e mapping on a surface of section. The continuous
flow in the $2n$-dimensional phase space is reduced to a discreet
flow in a $(2n-2)$-dimensional phase space, without loosing degrees
of freedom. Evidently, the Poincar\'e mapping of a Hamiltonian
system is symplectic. The difficulty is that it involves the solution
of the system, usually by numerical integrations.

Another method used widely is the method of averaging, which is especially
useful for the study of resonant motion. By this method, the "fast"
angles are eliminated and this results to a decrease of the
dimensions of the phase space, as with the Poincar\'e mapping. The
flow is also symplectic, but information is lost due to the
loss of one or more degrees of freedom.

The above methods are closely connected.  
Two methods have been developed recently  for the construction of
mapping models, for nearly integrable Hamiltonian systems of the form
$H_{0}+\epsilon H_{1}$, where $H_{0}$ in the integrable part.
Both are based on an averaged
Hamiltonian and though they are different, their basic philosophy is the
same: the lost degrees of freedom in the averaged model are regained, thus
recovering the information. 
 In the first method (Wisdom, 1983,85) the high frequency terms eliminated
in the averaging process are reintroduced, through periodic delta
functions, thus recovering the original degrees of freedom.
The solution obtained, considered at
integral multiples of a basic period, define the mapping. In the
second method (Hadjidemetriou 1991,93) the Poincar\'e mapping of the 
unperturbed Hamiltonian $H_{0}$ is obtained analytically and is
expressed through a generating function which is given
explicitly. Then the mapping is perturbed, by perturbing the
generating function, in such a way that the new mapping has all the essential
topological features of the original system. The perturbation is
based on the averaged Hamiltonian and the mapping is exactly
symplectic by its construction. In addition to these general methods,
some other symplectic maps are also presented, for particular cases
(e.g. cometary motion).

A basic problem related to the above mapping models is how good
is the averaged model. A simple test is presented to check the range
of validity of the averaged model and a method to correct it and
introduce any missing features (i.e. missing high eccentricity
resonances) is explained, even if the perturbation series do not
converge. The effects of the use of a wrong or incomplete
averaged model on the evolution of the system are discussed. 
 Since the properties of all similar dynamical
systems are essentially the same, as far as the evolution is
concerned, the different mapping models will have the same behaviour
and also the same properties as the original system,
provided that the topology of their phase space is the same. The
important point is to conserve the topology of the phase space of the
original system in constucting the mapping model.

We shall also show how a symplectic map is used to study the long term
evolution of a Hamiltonian system and give insight on the factors
that govern the evolution and on the mechanism of generation of chaos
through the interaction between the different degrees of freedom.

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