1549380323-Statistical Mechanics Theory and Molecular Simulation

26 Classical mechanics

ηi=xi−xi 0 , (1.7.3)

wherexi 0 −x(i+1)0=bi. The Hamiltonian in terms of the new variables and their
conjugate momenta is given by

H=

∑N

i=1

p^2 ηi

2 m

+

1

2

N∑− 1

i=1

mω^2 (ηi−ηi+1)^2. (1.7.4)

The equations of motion obeyed by this simple system can be obtaineddirectly from
Hamilton’s equations and take the form

η ̇i=

pηi

m

p ̇η 1 =−mω^2 (η 1 −η 2 )

p ̇ηi=−mω^2 (2ηi−ηi+1−ηi− 1 ), i= 2,...,N− 1

p ̇ηN=−mω^2 (ηN−ηN− 1 ), (1.7.5)

which can be expressed as second-order equations

̈η 1 =−ω^2 (η 1 −η 2 )

̈ηi=−ω^2 (2ηi−ηi+1−ηi− 1 ), i= 2,...,N− 1

η ̈N=−ω^2 (ηN−ηN− 1 ). (1.7.6)

In eqns. (1.7.5) and (1.7.6), it is understood that theη 0 =ηN+1= 0, since these
have no meaning in our system. Eqns. (1.7.6) must be solved subjectto a set of initial
conditions{η 1 (0),...,ηN(0),η ̇ 1 (0),...,η ̇N(0)}.
The general solution to eqns. (1.7.6) can be written in the form of a Fourier series

ηi(t) =

∑N

k=1

Ckaikeiωkt, (1.7.7)

whereωkis a set of frequencies,aikis a set of expansion coefficients, andCkis a
complex scale factor. Substitution of this ansatz into eqns. (1.7.6)gives

∑N

k=1

Ckωk^2 a 1 keiωkt=ω^2

∑N

k=1

Ckeiωkt(a 1 k−a 2 k)

∑N

k=1

Ckωk^2 aikeiωkt=ω^2

∑N

k=1

Ckeiωkt(2aik−ai+1,k−ai− 1 ,k)

∑N

k=1

Ckωk^2 aNkeiωkt=ω^2

∑N

k=1

Ckeiωkt(aNk−aN− 1 ,k). (1.7.8)

Since eqns. (1.7.8) must be satisfied independently for each function exp(iωkt), we
arrive at an eigenvalue equation of the form: