Introduction to Statistical Thermodynamics - Molecular Modelling - page 118

so that the total energy is

2 m d R

2

1 1

2 k s ( R

R e ) 2

ε = + −

d t

If I put q = R − R e then the momentum p is m d R /d t and I can write the Hamiltonian as

p 2
2 m +

k s q 2
2

H = (8.13)

We say that the particle moves through phase space and in this example the trajectory
through phase space is an ellipse (Figure 8.3), which can be easily seen by rewriting
Equation (8.13) as

p 2
a 2 +

q 2
b 2

= 1

q

p

Figure 8.3 Phase space

Hamilton's equations of motion are

d q i
d t =

∂ H
∂ p i ;

d p i
d t =−

∂ H
∂ q i

(8.14)

so for a general problemwith N atoms, we have to solve 6 N first-order differential equations
rather than the 3 N second-order differential equations we would get from straightforward
application of Newton's law.
In the case of a one-particle three-dimensional system, the Hamiltonian will be a function
of the three coordinates q and the three momenta p , and for a more general problem
involving N particles the Hamiltonian will be a function of the 3 N q 's and the 3 N p 's. We
say that the p 's and the q 's together determine a point in 6 N -dimensional phase space, and
this point is often denoted Γ .