23 April 2001
Physics Letters A 282 (2001) 276–283
Non-existence of the modiﬁed ﬁrst integral by symplectic
National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan
Received 2 February 2001; accepted 9 March 2001
Communicated by A.P. Fordy
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.
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)
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: email@example.com (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
H = H + τ
a differential polynomial of H . As a conse-
quence, the error of Hamiltonian remains as the or-
der of o(τ
) 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
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
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.