28 Classical mechanics
ζk(t) =ζk(0) cosωkt+pζk(0)
mωk
sinωkt k= 2,...,N, (1.7.15)whereζ 1 (0),...,ζN(0),pζ 1 (0),...,pζN(0) are the initial conditions on the normal mode
variables, obtainable by transformation of the initial conditions of the original coordi-
nates. Note thatpζ 1 (t) =pζ 1 (0) is the constant momentum of the free zero-frequency
mode.
In order to better understand the physical meaning of the normal modes, consider
the simple case ofN= 3. In this case, there are three normal mode frequencies given
by
ω 1 = 0, ω 2 =ω, ω 3 =
√
3 ω. (1.7.16)Moreover, the orthogonal transformation matrix is given by
U=
√^1
3√^1
2√^1
6√^1
3 0 −√^2
6√^1
3 −√^1
2√^1
6
. (1.7.17)
Therefore, the three normal mode variables corresponding to each of these frequencies
are given by
ω 1 = 0 : ζ 1 =1
√
3
(η 1 +η 2 +η 3 )ω 2 =ω: ζ 2 =1
√
2
(η 1 −η 3 )ω 3 =√
3 ω: ζ 3 =1
√
6
(η 1 − 2 η 2 +η 3 ). (1.7.18)These three modes are illustrated in Fig. 1.8. Again, the zero-frequency mode corre-
sponds to overall translations of the chain. Theω 2 mode corresponds to the motion
of the two outer particles in opposite directions, with the central particle remaining
fixed. This is known as theasymmetric stretchmode. The highest frequencyω 3 mode
corresponds to symmetric motion of the two outer particles with the central particle
oscillating out of phase with them. This is known as thesymmetric stretchmode. On
a final note, a more realistic model for real molecules should involve additional terms
beyond just the harmonic bond interactions of eqn. (1.7.1). Specifically, there should
be potential energy terms associated with bend angle and dihedralangle motion. For
now, we hope that this simple harmonic polymer model illustrates the types of tech-
niques used to solve classical problems. Indeed, the use of normalmodes as a method
for efficiently simulating the dynamics of biomolecules has been proposed (Sweetet al.,
2008). In general, for anharmonic potentials, one can only obtain local normal modes
obtained by diagonalizing the Hessian of the potentialHαβ=∂^2 U/∂qα∂qβin the
vicinity of a point in configuration space.