Computational Physics

8.4 Integration methods: symplectic integrators 219

and similar forH(δzb). This leads to the form

dδA

dt

∣

∣

∣∣

t= 0

=−(LTδza)·(Jδzb)−(δza)·(JLTδzb) (8.54)

whereLis the Jacobian matrix of the operatorJ∇zH:

Lij=

∑

k

Jik[∂^2 H(z)/∂zk∂zj]=

(

−Hpx −Hxx

Hpp Hpx

)

. (8.55)

HereHxxdenotes the second partial derivative with respect toxetcetera. It is easy
to see that the matrixLsatisfies

LTJ+JL=0, (8.56)

whereLTis the transpose ofL, and hence from (8.54) the areaδAis indeed
conserved.
We can now define symplecticity in mathematical terms. The Jacobi matrixSof
the mapping exp(tJ∇H)is given asS=exp(tL). This matrix satisfies the relation:

STJS=J. (8.57)

Matrices satisfying this requirement are calledsymplectic. They form a Lie group
whose Lie algebra is formed by the matricesLsatisfying(8.56). General nonlinear
operators are symplectic if their Jacobi matrix is symplectic.
In more than two dimensions the above analysis can be generalised foranypair
of canonical variablespi,xi– we say that phase space area is conserved for any
pair of one-dimensional conjugate variablespi,xi. The conservation law can be
formulated in an integral form[29]; this is depicted inFigure 8.3. In this picture the
three axes correspond top,xandt. If we consider the time evolution of the points
lying on a closed loop in thep,xplane, we obtain a tube which represents the flow
in phase space. The area conservation theorem says thatanyloop around the tube
encloses the same area

∮

pdx. In fact, there exists a similar conservation law for
volumes enclosed by the areas of pairs of canonical variables: these volumes are
called thePoincaré invariants. For the particular case of the volume enclosed by
areas ofallthe pairs of canonical variables, we recover Liouville’s theorem which
says that the volume in phase space is conserved. Phase space volume conservation
is equivalent to the Jacobi determinant of the time evolution operator in phase space
being equal to 1 (or−1 if the orientation is not preserved). For two-dimensional
matrices, the Jacobi determinant being equal to 1 is equivalent to symplecticity
as can easily be checked from(8.57). This is also obvious from the geometric
representation inFigure 8.3. For systems with a higher-dimensional phase space,
however, the symplectic symmetry is a more detailed requirement than mere phase
space conservation.