The Numerical Solution of Constrained Hamiltonian Systems.
Revised version appeared under the title Symplectic Integration of Constrained Hamiltonian Systems in: Mathematics of Computation, Vol. 63, No. 208 (1994) pp. 589-605
| SC 92-16 | B. Leimkuhler, S. Reich
The Numerical Solution of Constrained Hamiltonian Systems. Revised version appeared under the title Symplectic Integration of Constrained Hamiltonian Systems in: Mathematics of Computation, Vol. 63, No. 208 (1994) pp. 589-605 | |
Abstract: A Hamiltonian system subject to smooth
constraints can typically be viewed as a Hamiltonian system
on a manifold. Numerical computations, however, must be
performed in 
. In this paper, canonical
transformations from ``Hamiltonian differential-algebraic
equations to ODEs in Euclidean space are considered.
In
§2, canonical parameterizations or local charts are
developed and it is shown how these can be computed in a
practical framework. In §3 we consider the construction of
unconstrained Hamiltonian ODE systems in the space in which
the constraint manifold is embedded which preserve the
constraint manifold as an integral invariant and whose flow
reduces to the flow of the constrained system along the
manifold. It is shown that certain of these unconstrained
Hamiltonian systems force Lyapunov stability of the
constraint-invariants, while others lead to an unstable
invariant. In §4, we compare various projection techniques
which might be incorporated to better insure preservation of
the constraint-invariants in the context of numerical
discretization. Numerical experiments illustrate the degree
to which the constraint and symplectic invariants are
maintained under discretization of various formulations.
Keywords: differential-algebraic equations,
Hamiltonian systems, canonical discretization schemes.
AMS(MOS): subject classification 65L05.
Keywords: differential algebraic equations,
Hamiltonian system,
canonical discretization schemes
MSC: 65L05