FORMAL LIE SERIES AND APPLICATIONS TO THE HAMILTONIAN
MECHANICS
Date: July 19th (Friday)
Time: ??.??
Abstract
We consider different applications of formal Lie series to the hamiltonian mechanics. First
example is the symplectic integration. Construction of most known symplectic methods, like
Yoshida method is based on iterating process of superpositions of the exponential maps which can
be represented as Lie series. Usually such superpositions can be evaluated on the base of BCH
formula. We show how to construct implicit symplectic methods for numerical integration of
hamiltonian ODE if the hamiltonian does not have the separated form
.
Another problem which is also connected with application of
symplectic methods is calculating of perturbation of hamiltonian due to concrete symplectic
method for numerical integration. Some results about modeling of Henon-Heyles system via
symplectic and nonsymplectic methods are also presented. We show how to use the procedure of
normalisation based on the Lie series in order to construct the formal integral for the symplectic
approximations of the Henon-Heiles system. We also discribe some algorithms for processing
with formal Lie series, the problem of computer constructing of special basses in free Lie algebra
and methods for representation of Lie series in the computer memory which allow effectively
realize such algorithms.
