396 J. E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY
and McCracken [1978] and Abraham, Marsden and Ratiu [1988] for details. An
example of this is the {Lie-Trotter) product formula
e t(A+B) = lim (etA/netBfn)n (5.3)
n->oowhich is a time splitting method for solving :i; = Ax + Bx by iterat ing, in an
alternating fashion, the solutions of :i; = Ax and :i; = Bx.
An algorithm FT is
- a symplectic integrator if each FT is symplectic,
2. an energy integrator if Ho FT= H (where X = XH),
3. a momentum integrator if J o FT = J , where J is the momentum map
for the action of a Lie group G.If FT has one or more of t hese properties , we call it a mechanical integrator.
Notice that if an integrator has one of these three properties, then so does any
iterate of it, since these properties are preserved by composition.
There are several ways that have b ee n employed t o find mechanical integrators.
For example, one can search amongst existing algorithms and find ones with special
algebraic properties that make them symplectic or energy-preserving. Alternatively,
one can attempt to design mechanical integrators from scratch. Here are some
simple examples:
Example 1. A first order explicit symplectic scheme in the plane is given by the
map (qo,Po) f-+ (q,p) defined byq qo + (D.t)po
p = Po - (D.t)V'(qo + (D.t)po). (5.4)
This map is a first order approximation to the flow of Hamilton's equations for the
Hamiltonia n H = (p^2 /2) + V(q). Here, one can verify by direct calculation that
this scheme is symplectic. +
Example 2. An implicit, symplectic, second order accurate scheme in the plane
for the same Hamiltonian as in Example 1 is
q
pqo + (D.t)(p + Po)/2
Po - (D.t)V'((q + qo)/2). + (5.5)
Shortly we shall see how to construct such algorithms systematically, but for
now just regard t hem as illustrat ive. The next example shows that the second order
accurate mid-point rule is symplectic (Feng [1986]). This algorithm is also useful in
developing almost Poisson integrators (Austin , Krishnaprasad and Wang [1993]).
Example 3. In a symplectic vector space t he following mid point rule for a
Hamiltonian vector field XH is symplectic:
zk+l _ zk _ ( z k + zk+l)
Dot - XH 2. (5.6)Notice that for small Dot the map given implicit ly by this equation is well defined
by the implicit function theorem. One way to show that it is symplectic, is to use
the fact that the Cayley transform S of an infinitesimally symplectic linear map
A, namely