Extended phase space 183dri
dt′=
p′i
midp′i
dt′=Fi−sp′s
Qp′ids
dt′=
s^2 p′s
Qdp′s
dt′=
1
s[N
∑
i=1(p′i)^2
mi−gkT]
−
s(p′s)^2
Q. (4.8.12)
Because of the noncanonical transformation, these equations lose their symplectic
structure, meaning that they are no longer incompressible. In addition, they involve
an unconventional definition of time due to the scaling by the variables. This scaling
makes the equations somewhat cumbersome to use directly in the form of (4.8.12). In
the next few sections, we will examine two methods for transforming the Nos ́e equa-
tions into a form that is better suited for use in molecular dynamics calculations.
4.8.2 The Nos ́e-Poincar ́e Hamiltonian
The Nos ́e–Poincar ́e method (Bondet al., 1999) is named for a class of transformations
known asPoincar ́e transformations, which are time-scaling transformations commonly
used in celestial mechanics (Zare and Szebehely, 1975). Given a HamiltonianH(x), we
define a transformed HamiltonianH ̃(x) by
H ̃(x) =f(x)(
H(x)−H(0))
, (4.8.13)
whereH(0)is the initial value of the HamiltonianH(x), andf(x) is an arbitrary scalar
function of x. The equations of motion derived fromH ̃(x) are
̇x =f(x)M∂H
∂x+
(
H(x)−H(0))
M
∂f
∂x, (4.8.14)
where M is the matrix in eqn. (1.6.27). Eqn. (4.8.14) shows that whenH(x) =H(0),
the equations of motion are related to the usual Hamiltonian equations dx/dt′ =
M(∂H/∂x) by the time scaling transformation dt′= dt/f(x).
Bond,et al.(1999) exploited this type of transformation to yield a new ther-
mostatting scheme with the correct intrinsic definition of time. Based on our analysis
of the Nos ́e Hamiltonian, it is clear that to “undo” the time scaling, weshould choose
f(x) =sand define a transformed Hamiltonian