Computational Physics

8.4 Integration methods: symplectic integrators 221

increasing order of the integrator. Later Yoshida[21]and Forest[22]developed a
different scheme for finding symplectic integrators, and in this section we follow
their analysis.
Consider a Hamiltonian of the simple form:
H=T(p)+U(x) (8.58)

(we still restrict ourselves to a particle in one dimension – results are easily
generalised). In terms of the variablez=(p,x)the equations of motion read

dz

dt

=

(

−

∂H

∂x

,

∂H

∂p

)

=

(

−

∂U(x)

∂x

,

∂T(p)

∂p

)

=J∇H(z)≡T ̃(z)+U ̃(z), (8.59)

where in the last expression the operatorJ∇H, which acts onz=(p,x), is split
into the contributions from the kinetic and potential energy respectively:

T ̃(z)=

(

0,

∂T(p)

∂p

)

(8.60a)

U ̃(z)=

(

−

∂U(x)

∂x

,0

)

. (8.60b)

T ̃andU ̃are therefore also operators which map a pointz=(p,x)in phase space
onto another point in phase space.
As we have seen in the previous section, the exact solution of(8.59)is given as
z(t)=exp(tJ∇H)[z( 0 )]=exp[t(T ̃+U ̃)][z( 0 )]. (8.61)

The term exp(tJ∇H)is a time evolution operator. It is a symplectic operator, as
are exp(tTˆ)and exp(tUˆ)since these can both be derived from a Hamiltonian (for
a free particle and a particle with infinite mass respectively).
An nth order integratorfor time stephis now defined by a set of numbersak,bk,
k=1,...,m, such that

∏m

k= 1

exp(akhT ̃)exp(bkhU ̃)=exp(hJ∇H)+O(hn+^1 ). (8.62)

Since the operators exp(akhT ̃)and exp(bkhU ̃)are symplectic, the integrator (8.62)
is symplectic too. The difference between the integrator and the exact evolution
operator can be expressed in Campbell–Baker–Hausdorff (CBH) commutators: if
eC=eAeBthen

C=A+B+[A,B]/ 2 +([A,[A,B]]+[B,[B,A]])/ 12 +··· (8.63)

where the dots represent higher order commutators. This formula can be derived
by writing exp(tA)exp(tB)=exp[t(A+B)+], expanding the operatorin