Time step dependent Hamiltonians 123pxFig. 3.4Phase space plot of the shadow Hamiltonian in eqn. (3.13.3) for different time steps.
The eccentricity of the ellipse increases as the time step increases.
which is known as the Baker–Campbell–Hausdorff formula, connectsthe two propaga-
tors (see, for example, Yoshida, 1990). Here, the operatorsCkare nested commutators
of the operatorsiL 1 andiL 2. For example, the operatorC 1 is
C 1 =−
1
24
[iL 2 + 2iL 1 ,[iL 2 ,iL 1 ]]. (3.13.5)Now, an important property of the Liouville operator for Hamiltoniansystems is that
such commutators as those in eqn. (3.13.5) yield new Liouville operators that cor-
respond to Hamiltonians derived from analogous expressions involving the Poisson
bracket. Consider, for example, the simple commutator [iL 1 ,iL 2 ]≡ −iL 3. It is pos-
sible to show thatiL 3 is derived from the HamiltoniansH 1 (x) andH 2 (x), which
defineiL 1 andiL 2 , respectively, viaiL 3 ={...,H 3 }, whereH 3 (x) ={H 1 (x),H 2 (x)}.
The proof of this is straightforward and relies on an important identity satisfied by
the Poisson bracket known as theJacobi identity: IfP(x),Q(x), andR(x) are three
functions on the phase space, then the Jacobi identity states
{P,{Q,R}}+{R,{P,Q}}+{Q,{R,P}}= 0. (3.13.6)Note that the second and third terms are generated from the first by moving the
functions around in a cyclic manner. Thus, consider the action of [iL 1 ,iL 2 ] on an
arbitrary phase space functionF(x). SinceiL 1 ={...,H 1 (x)}andiL 2 ={...,H 2 (x)},
we have
[iL 1 ,iL 2 ]F(x) ={{F(x),H 2 (x)},H 1 (x)}−{{F(x),H 1 (x)},H 2 (x)}. (3.13.7)