218 Molecular dynamics simulations
It is convenient to introduce the combined momentum–position coordinatez =
(p,x), in terms of which the equations of motion read
̇z=J∇H(z) (8.46)whereJis the matrix
J=(
0 − 1
10
)
(8.47)
and∇H(z)=(∂H(z)/∂p,∂H(z)/∂x).^5
Expanding the equation of motion(8.46)to first order, we obtain the time
evolution of the pointzto a new point in phase space:
z(t+h)=z(t)+hJ∇zH[z(t)]. (8.49)The exact solution of the equations of motion can formally be written as
z(t)=exp(tJ∇zH)[z( 0 )] (8.50)where the exponent is to be read as a series expansion of the operatortJ∇zH.
This can be verified by substituting Eq. (8.50) into (8.46). This is a one-parameter
family of mappings with the timetas the continuous parameter. The first order
approximation to (8.50) coincides with (8.49).
Now consider a small region in phase space located atz=(p,x)and spanned by
the infinitesimal vectorsδzaandδzb. The areaδAof this region can be evaluated
as the cross product ofδzaandδzbwhich can be rewritten as^6
δA=δza×δzb=δza·(Jδzb). (8.51)It is now easy to see that the mapping(8.50)preserves the areaδA. It is sufficient
to show that its time derivative vanishes fort=0, as for later times the analysis
can be translated to this case. We have
dδA
dt∣
∣∣
∣
t= 0=
d
dt{[etJ∇zH(δza)]·[JetJ∇zH(δzb)]}t= 0=[J∇zH(δza)]·(Jδzb)+(δza)·[JJ∇zH(δzb)]. (8.52)We can findH(δza,b)using a first order Taylor expansion:
H(δza)=H(z+δza)−H(z)=δza·∇zH(z), (8.53)(^5) In more than one dimension, the vectorzis defined as(p 1 ,...,pN,x 1 ,...,xN), and the matrixJreads in
that case
J=
( 0 −I
I 0
)
(8.48)
whereIis theN×Nunit matrix.
(^6) Note that the area can be negative: it is anorientedarea. In the language of differential geometry this area
is called atwo-form.