Problems 511.6. A water molecule H 2 O is subject to an external potential. Let the positions
of the three atoms be denotedrO,rH 1 ,rH 2 , so that the forces on the three
atoms can be denotedFO,FH 1 , andFH 2. Consider treating the molecule
as completely rigid, with internal bond lengthsdOHanddHH, so that the
constraints are
|rO−rH 1 |^2 −d^2 OH= 0
|rO−rH 2 |^2 −d^2 OH= 0
|rH 1 −rH 2 |^2 −d^2 HH= 0.a. Derive the constrained equations of motion for the three atomsin the
molecule in terms of undetermined Lagrange multipliers.
b. Show that the forces of constraint do not contribute to the work done on
the molecule in moving it from one spatial location to another.
c. Determine Euler’s equations of motion about an axis perpendicularto
the plane of the molecule in a body-fixed frame whose origin is located
on the oxygen atom.
d. Determine the equations of motion for the quaternions that describe this
system.1.7. Calculate the classical action for a one-dimensional free particle of massm.
Repeat for a harmonic oscillator of spring constantk.
1.8. A simple mechanical model of a diatomic molecule bound to a flat surface
is illustrated in Fig. 1.13. Suppose the atom with massesm 1 andm 2 carry
electrical chargesq 1 andq 2 , respectively, and suppose that the molecule is
subject to a constant external electric fieldEin the vertical direction, directed
upwards. In this case, the potential energy of each atom will beqiEhi,i= 1, 2
wherehiis the height of the atomiabove the surface.
a. Usingθ 1 andθ 2 as generalized coordinates, write down the Lagrangian
of the system.
b. Derive the equations of motion for these coordinates.
c. Introduce the small-angle approximation, which assumes that the angles
only execute small amplitude motion. What form do the equations of
motion take in this approximation?
1.9. Use Gauss’s principle of least constraint to determine a set of non-Hamiltonian
equations of motion for the two atoms in a rigid diatomic molecule of bond
lengthdsubject to an external potential. Take the constraint to be
σ(r 1 ,r 2 ) =|r 1 −r 2 |^2 −d^2.Determine the compressibility of your equations of motion.