52 Classical mechanics
(^1)
q 2
q
l 2
l 1
m 1
m 2
Fig. 1.13Schematic of a diatomic molecule bound to a flat surface.
1.10. Consider the harmonic polymer model of Section 1.7 in which the harmonic
neighbor couplings all have the same frequencyωbut the masses have al-
ternating valuesmandM, respectively. For the case ofN = 5 particles,
determine the normal modes and their associated frequencies.
1.11. The equilibrium configuration of a molecule is represented by three atoms
of equal mass at the vertices of a right isosceles triangle. The atoms can be
viewed as connected by harmonic springs of equal force constant. Find the
normal mode frequencies of this molecule, and, in particular, show that the
zero-frequency mode is triply degenerate.
1.12. A particle of massmmoves in a double-well potential of the form
U(x) =
U 0
a^4(
x^2 −a^2) 2
.
Sketch the contours of the constant-energy surface H(x,p) =E in phase
space for the following cases:
a.E < U 0.
b. E=U 0 +ǫ, whereǫ≪U 0.
c. E > U 0.∗1.13. The Hamiltonian for a system ofNcharged particles with chargesqi,i=
1 ,...,Nand massesmi,i= 1,...,N, positions,r 1 ,...,rNand momentap 1 ,...,pN