23 April 2001
Physics Letters A 282 (2001) 276–283
www.elsevier.nl/locate/pla
Non-existence of the modified first integral by symplectic
integration methods
Haruo Yoshida
National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan
Received 2 February 2001; accepted 9 March 2001
Communicated by A.P. Fordy
Abstract
It is known that the symplectic mapping obtained as a symplectic integration method is formally an exact time evolution of the
modified Hamiltonian which is close to the original Hamiltonian. In the case when the original Hamiltonian has an additional
first integral, it is shown that the modified first integral, which is defined to be an integral for the modified Hamiltonian, does not
necessarily exist in general. This non-existence of the modified first integral is demonstrated by an example of the 2D harmonic
oscillator with an integer frequency ratio.
2001 Elsevier Science B.V. All rights reserved.
1. Introduction
Symplectic integration method (symplectic integra-
tor) is a numerical integration method for Hamiltonian
systems which is designed to conserve the property
that the method (mapping)
(1)
q(t),p(t)
→
q(t + τ),p(t + τ)
for one step, τ = t, is exactly symplectic as the
original Hamiltonian flow. It is known that for general
Hamiltonian systems, implicit symplectic methods
such as implicit Runge–Kutta methods always exist.
On the other hand, for a Hamiltonian of the form H =
T(p)+ V(q), explicit methods are easily constructed
[6,10]. For a review, see [4,8,11].
When a symplectic method is applied to a concrete
example, it is observed that although the value of
Hamiltonian is not conserved at each step, the error
does not grow monotonically and remains in some
E-mail address: h.yoshida@nao.ac.jp (H. Yoshida).
finite width. This phenomena can be explained naively
by the fact that any symplectic method is a rigorous
time-τ evolution of the modified Hamiltonian,which
is close to the original Hamiltonian [2,11].
For an nth-order symplectic method, this modified
Hamiltonian takes the form
(2)
˜
H = H + τ
n
H
n
+···
with H
n
a differential polynomial of H . As a conse-
quence, the error of Hamiltonian remains as the or-
der of o(τ
n
) forever, if the series converges and if the
round-off error is neglected. However, since this se-
ries diverges in general, any rigorous statement can be
claimed only after a truncation of the series at some fi-
nite order. As a result of the truncation, it turns out that
the error of Hamiltonian remains as the order of o(τ
n
)
for an exponentially long time interval [1,3].
Suppose now that the given Hamiltonian system
is integrable and that there exists a first integral
(integral of motion, conserved quantity) besides the
Hamiltonian. Then there is no guarantee that this
first integral is also well conserved by the symplectic
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