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\begin{center}{\large\bf Nonconcervative effects of the }\end{center}
\begin{center}{\large\bf numerical modeling of Hamiltonian systems }\end{center}
\begin{center}{\large\bf and symplectic integration}\end{center}

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\begin{center}{\bf Nickolay N. Vasiliev}\end{center}

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\begin{center}{\it Institute of Theoretical Astronomy of Russian Academy of Scie
nces \\ }\end{center}
\begin{center}{\it ITA RAS, Nab. Kutuzova, 10, St.Petersburg, 191187, Russia \\ 
}\end{center}

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This report devoted to the influence of nonconservative effects of the
numerical modeling methods to the topology of phase portraits of
systems of Hamiltonian differential equations. Also we study exact symplectic
methods for numerical integration of Hamiltonian systems.

Usually process of modeling of hamiltonian systems is based on some methods of
numerical integration of ODE .  Orthodox methods like RK methods are not
symplectic therefore they do not conserve special properties of
Hamiltonian systems like volume preservation.
The application of orthodox methods of modeling
leads to global changes of topology of the phase portrait. For example
attractors can appear. This effect usually appears at integration over a very
long time span, as the accuracy of calculations of individual
trajectories can be very high.  Not long ago exact symplectic
methods for numerical integration of Hamiltonian differential
equations appeared [1]. They do not bring nonconservative
perturbations into the modeled systems. Therefore they keep the
properties of the phase portrait, which are stable relative to
the small conservative perturbations.

We present a scheme to study of nonconservative effects of
orthodox numerical modeling methods, which is based on applications of
computer algebra. We construct for a concrete
Hamiltonian the one-step map for the method of integration in exact
symbolic form.
Jacobian of this map is not equal~1, as the Liouville's theorem does
not hold for orthodox methods. Therefore the phase space consists of
domains of compression and domains of expansion. The behavior of the
families of trajectories of dynamical systems, which fall in the one
from these domains, strongly changes at the process of modeling. The
analytical calculations of the Jacobian of the one-step map can be
made by means of computer algebra system like Reduce and Maple.
in analytical form too. Solving the equation det$(J)=1$ we get a
manifold, which separates the domains of expansion from ones of
compression.
As an example the restricted three body problem and the Henon-Heiles
Hamiltonian of two degrees of freedom
were considered. To get the phase portrait a Poincare map was
constructed. We studied the nonconservative effects of the
Runge-Kutta method near libration points for restricted
three body problem and in all phase space for Henon-Heyles hamiltonian.
As energy dissipation in observed
we could not limit oneself by investigation of the phase portrait on
the standard section of Poincare map only, where E is constant on this
standard section. Therefore the phase portraits were presented at
other sections, where first coordinate means energy E and second one is
a function of coordinates and impulses of the system.

It will be noted that an integration  by exact
symplectic method allows to keep a topology of
the phase portrait almost on the whole phase space.
We also present an explicit symplectic scheme for the elliptic
restricted three body problem. In this case
hamiltonian has not separate form and we can not directly
use Yoshida methods, but it can be
presented as sum of two integrable hamiltonians.
Beside this we present some results about estimation of errors of
action variables for symplectic methods. These estimations are based
on KAM theory.
At the end we discuss some applications of manipulating with formal
Lie series to the problem of constructing symplectic integrators.

\begin{center}{\bf References }\end{center}

1. H.Kinoshita, H.Yoshida and H.Akai. {\it Symplectic integrators and their
application to dynamical astronomy.} Celest. Mech. and Dyn. Astr. v.50,
59-71, 1991.

2. N.N.Vasiliev and N.A.Pavlova. {\it Exact symplectic methods of modeling
of dynamical systems.} In "Report of Russian Conference on Computer
Methods of Celestial Mechanics", 15-16, St.Petersburg, 1992, (in
Russian).

3. N.A.Sushko., N.N.Vasiliev,
{\it Study of Nonconservative Effects of the Numerical Modeling of
Hamiltonian Systems by Means of Computer Algebra} New Computer Technologies
in Control Systems. Proceedings of the International Workshop.
Perelavl-Zalessky, 1994.
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