Goldstein

1. Lecture - 5
Oscillations
 Formulation of the Problem
 The Eigenvalue Equation
 Frequencies of Free Vibration, and Normal Coordinates
 Free Vibrations of a Linear Triatomic Molecule
 Forced Vibrations and the Effect of Dissipative Forces
 The Damped Driven Pendulum

2. Small Oscillations
• For conservative systems in which the potential energy is a function of position only.
• Assumed that the transformation equations defining the generalized coordinates of the system,
q1, . . . , qn, do not involve the time explicitly. Thus, time-dependent constraints are to be
excluded.
• The system is said to be in equilibrium when the generalized forces acting on the system
vanish:
• The potential energy has an extremum at the equilibrium configuration of the system, q01, q02, .
. . , q0n.
• If the configuration is initially at the equilibrium position, with zero initial velocities qn , then the
system will continue in equilibrium indefinitely.

3. . . . . Continue . . . . . . . .
• An equilibrium position is classified as stable if a small disturbance of the
system from equilibrium results only in small bounded motion about the rest
position.

4. . . . . Continue . . . . . . . .
• An equilibrium position is classified as stable if a small disturbance of the
system from equilibrium results only in small bounded motion about the rest
position.
• The equilibrium is unstable if an infinitesimal disturbance eventually
produces unbounded motion.
• A pendulum at rest is in stable equilibrium, but the egg standing on end is an
obvious illustration of unstable equilibrium.
• It can be readily seen that when the extremum of V is a minimum the
equilibrium must be stable.
FIGURE 6.1 Shape of the potential
energy
curve at equilibrium.
• It can be readily seen that when the extremum of V is a
minimum the equilibrium must be stable.

5. . . . . Continue . . . . .
Consider the motion in the neighborhood of a configuration of stable equilibrium. Since the departures from
equilibrium are too small, all functions may be expanded in a Taylor series about the equilibrium, retaining only
the lowest-order terms.
• The deviations of the generalized coordinates from equilibrium will be denoted by ηi :
These may be taken as the new generalized coordinates of the
motion.
Expanding the potential energy about q0i , we obtain

6. . . . . . Continue . . . .
• The first term in the series is the potential energy of the equilibrium position, and by shifting the arbitrary zero
of potential to coincide with the equilibrium potential, this term may be made to vanish.
• The terms linear in ηi also vanish automatically in consequence of the equilibrium conditions, we are
expanding about the minimum.
We are therefore left with the quadratic terms as the first approximation to V:
where the second derivatives of V have been designated by Vij depending only upon the equilibrium values of
the qi ’s. It is obvious from their definition that the Vij ’s are symmetrical, that is, Vij = Vji.
Thus, the potential can simply be independent of a particular coordinate, so that equilibrium occurs at any
arbitrary value of that coordinate. We speak of such cases as neutral equilibrium.

7. … Continue …..
Now for the kinetic energy. In
terms of generalized
coordinates,
Since the generalized coordinates do not involve the time explicitly,
the kinetic energy is a homogeneous quadratic function of the
velocities;
Denoting the constant values of the mi j functions at equilibrium by Ti j (Inertia of system), we can therefore
write the kinetic energy as;
⸫Diff. eq 6.2

8. . . . . . Continue . . . .
Ti j must be symmetric, since the individual terms in Eq. (6.6) are unaffected by an interchange of indices. From
Eqs. (6.4) and (6.6), the Lagrangian is given by;
Using Lagrange formalism equations of motion are;
the kinetic energy term can be easily written so as to have no cross terms, corresponds to the Lagrangian;
which generates the equations of motion;
which generates the equations of motion;

9. 2. The Eigenvalue Equation
The equations of motion are 2nd order diff. equations, these eqs. describe the
motion of the system near the equilibrium. These equations are same as for 1-D harmonic oscillators, so for ηi
(an oscillatory solution) is;
Here Cai gives the complex amplitude of the oscillation for each coordinate ηi , the factor C being introduced for
convenience as a scale factor, the same for all coordinates. The real part of Eq. (6.11) is to correspond to the
actual motion. Substitution of the trial solution (6.11) into the equations of motion leads to the following equations;
Equations (6.12) provide n linear homogeneous equations for the ai ’s, and consequently can have a nontrivial solution
only if the determinant of the coefficients vanishes:

10. • This is in effect an algebraic equation of the nth degree for 𝜔 𝛼
2
, where; α = 1,2,…… n.
• the roots of the determinant provide the frequencies for which Eq. (6.11) represents a solution to the equations
of motion.
• For each of these values of 𝜔 𝛼
2
, eqs. (6.12) may be solved for the amplitudes of aiEquations (6.12) represent a type of eigenvalue equation, for writing Ti j as an element of the matrix T, the
equations may be written
This is an eigenvalue equation.
V acting on eigenvector a produces a multiple of the result of T acting on a. The eigenvalues λ for which Eq. (6.14) all
real in consequence of the symmetric and reality properties of T and V, and, in fact, must be positive.

11. Here to determine the amplitude of the perturbation along generalized coordinates qi in mode α.
Use equations of motion;
(Vij - 𝜔 𝛼
2
Tij) Ajα = 0 ⸫Fixed value of α.
For a known 𝝎α , these are n equations in the amplitude vector.
• 𝜔 𝛼
2 must be positive and real.
• Assume that all 𝝎α are known, must now determine the amplitude ai.
• Determination of amplitude means to know what parts of the system are moving and by how much are
they moving in the oscillation mode α.
• Patterns of motion in which all parts of the system move sinusoidally with same frequency and with fixed
phase relation.

12. 3. Frequencies of Free Vibration, and Normal Coordinates
The equations of motion will be satisfied by an oscillatory solution of the form (6.11), not only for one frequency
but in general for a set of n frequencies ωk .
• A complete solution of the equations of motion involves a superposition of oscillations with all the allowed
frequencies.
• If the system is displaced slightly from equilibrium and then released, the system performs small oscillations
about the equilibrium with the frequencies ω1, . . . , ωn.
• The solutions of these equations are therefore often designated as the frequencies of free vibration or as the
resonant frequencies of the system.
• The general solution of the equations of motion may now be written as a summation over an index k:

13. There is a complex scale factor Ck for each resonant frequency.
It might be objected that for each solution λk of the equation there are two resonant frequencies +ωk and −ωk .
The eigenvector ak would be the same for the two frequencies, but the scale factors 𝐶 𝑘
+
and 𝐶 𝑘
−
could
conceivably be different. On this basis, the general solution should appear as;
Recall however that the actual motion is the real part of the complex solution, and the real part of either (6.35)
or (6.35´) can be written in the form;
where the amplitude fk and the phase δk are determined from the initial conditions. Either of the solutions ((6.35)
and (6.36)) will therefore represent the actual motion.

14. • The solution for each coordinate, Eq. (6.35), is in general a sum of simple harmonic oscillations in all of the
frequencies ωk satisfying the equation of motion.
• ηi never repeats its initial value and is therefore not itself a periodic function of time. However, it is possible to
transform from the ηi to a new set of generalized coordinates that are all simple periodic functions of time—a
set of variables known as the normal coordinates. We define a new set of coordinates ζ j
or, in terms of single-column matrices η and ζ,
The potential energy, Eq. (6.4), is written in matrix notation as;
Now, the single-row transpose matrix is related to by the
equation
ζ𝜂

15. so that the potential energy can be written also as
But A diagonalizes V by transformation (Eq. (6.26)), and the potential energy therefore reduces simply to
The kinetic energy has an even simpler form in the new coordinates. Since the velocities transform as the
coordinates, T as given in Eq. (6.20) transforms to
which by virtue of Eq. (6.23) reduces to,
The new Lagrangian is
so that the Lagrange equations for ζk are,

16. Equations (6.46) have the immediate solutions,
• Each of the new coordinates is thus a periodic function involving only one of the resonant frequencies.
• The ζ ’s the normal coordinates of the system.
• Each normal coordinate corresponds to a vibration of the system with only one frequency, and these
component oscillations are called the normal modes of vibration.

17. 4. Free Vibrations of a Linear Triatomic Molecule
• In the equilibrium configuration of the molecule, two atoms of mass m are symmetrically located on each side
of an atom of mass M (Fig. 6.3). All three atoms are on one straight line, the equilibrium distances apart being
denoted by b.
• First consider only vibrations along the line of the molecule, and the actual interatomic potential will be
approximated by two springs of force constant k joining the three atoms.
• There are three coordinates marking the position of the three atoms on the line. In these coordinates,
potential energy is,

18. We now introduce coordinates relative to the equilibrium
positions:
Where;
The potential energy then reduces to
or
The V tensor has the form;

19. The kinetic energy has an even simpler form:
The T tensor is:
Combining these two tensors, the equation appears as;
Direct evaluation of the determinant leads to the cubic equation in ω2:

20. with the solutions;
The first eigenvalue, ω1 = 0. Such a solution does not correspond to an oscillatory motion at all, for the equation
of motion for the corresponding normal coordinate is which produces a uniform translational motion.
The vanishing frequency arises from the fact that the molecule may be translated rigidly along its axis without any
change in the potential energy, is an example of neutral equilibrium. Since the restoring force against such motion is zero,
the effective “frequency” must also vanish.
ω2 is the frequency of oscillation for a mass m suspended by a spring of force constant k. We are therefore led
to expect that only the end atoms partake in this vibration; the center molecule remains stationary.
It is only in the third mode of vibration, ω3, that the mass M can participate in the oscillatory motion. These
predictions are verified by examining the eigenvectors for the three normal modes.