2. Chapter 6: Oscillations Sect. 6.1: Formulation of Problem
Conservative systems
 PE V is a function of position only.
Assume: The transformation eqtns defining generalized coords q1,q2, ,,,qn, do not explicitly involve the time.  No time dependent constraints.
If the system is in Equilibrium  All generalized forces Qi vanish  Qi = (V/qi)0 = 0 (all i)
 V has an extremum at the equilibrium
configuration:
q1,q2, ,,,qn, = q01,q02, ,,,q0n,

3. Stable Equilibrium  A small disturbance from equilibrium results in small bounded motion around the equilibrium position.  At the equilibrium position qi, the extremum of V is a minimum.
 (2V/qiqj)0 > 0 (See figure)
Describing the motion
which occurs with such
a small disturbance
away from stable
equilibrium is the
subject of this chapter!

4. Unstable Equilibrium  A small disturbance from equilibrium results in unbounded motion.
 At the equilibrium position qi, the extremum of V is a maximum.  (2V/qiqj)0 < 0 (See figure)

5. V(q1,q2, ,,,qn)  V(q01,q02, ,,,q0n) + (V/qi)0ηi
We’re interested in the case where the system is initially in stable equilibrium & is slightly disturbed away from it.
 Small departures from equilibrium  Expand V in a Taylor’s series about equilibrium, keep only lowest order terms. Write: qi = q0i + ηi : Expand V about q0i & keep lowest order terms in ηi (summation convention):
V(q1,q2, ,,,qn)  V(q01,q02, ,,,q0n) + (V/qi)0ηi
+ (½)(2V/qiqj)0ηiηj + ...
Note: (V/qi)0 = 0 by the equilibrium conditions.
V(q01,q02, ,,,q0n) = a constant. Does not affect the motion. By shifting the (arbitrary) zero of the PE, we can set it to zero.

6. Drop terms higher than quadratic in the displacements from equilibrium ηi
 V(q1,q2, ,,,qn) = (½)(2V/qiqj)0ηiηj
Define: Vij  (2V/qiqj)0
 V= (½)Vijηiηj
Note:
Vij are symmetric: Vij = Vji .
It is possible for Vij to vanish Vij = 0
 Neutral Equilibrium
In what follows, we assume Vij  0

7. Generalized coords do not contain time explicitly:
Now, consider the KE, T:
Generalized coords do not contain time explicitly:
 T is a homogeneous, quadratic function of the generalized velocities qi.
This is a theorem proven in Ch. 1!
 Can write: T = (½)mijqiqj = (½)mijηiηj
Coefficients mij: Not simply masses! Involve masses & transformation eqtns between Cartesian coords & generalized coords. In general, functions of generalized coords. See Ch. 1! Can expand about equilib in similar fashion to V:
mij = mij(q1 ,,,qn)  mij(q01,,,q0n) + [(mij)/qk]0ηk +…
Keep only the constant term & define: Tij  mij(q01,,,q0n)

8.  Tijηj + Vijηj = 0 (i = 1,2,3, … n) (3)
 T = (½)Tijηiηj
Note:
Tij are symmetric: Tij = Tji
Now, consider Lagrangian for system slightly displaced from stable equilibrium:
L = T - V = (½)(Tijηiηj - Vijηiηj) (1)
Lagrange Eqtns of Motion, using (1):
(d/dt)[(L/ηi)] - (L/ηi) = 0 (i = 1,2,3, … n) (2)
 Tijηj + Vijηj = 0 (i = 1,2,3, … n) (3)
n coupled (simple harmonic oscillator) equations of motion!

9. In many common cases, T has no cross terms. That is Tij = Tiδij
 T = (½)Ti(ηi )2
So, in this case, the Lagrangian is:
L = T - V = (½)[Ti(ηi)2 - Vijηiηj]
Lagrange Eqtns of Motion become:
 Tiηi + Vijηj = 0 (i = 1,2,3, … n)
No sum on i.

10. Sect. 6.2: Eigenvalue Equation & Principal Axis Transformation
We want to solve the equations of motion for a system slightly perturbed from stable equilibrium:
Tijηj + Vijηj = 0 (i = 1,2,3, … n) (3)
(3): n linear (coupled simple harmonic oscillator) differential equations, n unknowns: Try (complex) oscillatory solutions:
ηj  Caje-iωt
The real part corresponds to the physical motion
Substituting this into (3)
 (Vijaj - ω2Tijaj ) = (i = 1,2,3, … n) (4)
(4): n homogeneous linear algebraic equations, n unknowns (aj):
 The determinant of the coefficients of the aj vanishes!

11. (Vijaj - ω2Tijaj ) = 0 (i = 1,2,3, … n) (4)
That is:
V11 - ω2T V12 - ω2T12 V13 - ω2T13 …..
V21 - ω2T V22 - ω2T22 V23 - ω2T23 … = (5)
V31 - ω2T V32 - ω2T32 V33 - ω2T33 …..
…………..
(5): An nth degree algebraic equation for ω2: These roots are the allowed frequencies with which the system can vibrate (or oscillate). Once (5) is solved for the n ω2  (ω2)k for each solution, can go back to
(Vijaj - ω2Tijaj ) = (i = 1,2,3, … n) (4)
and solve for the amplitudes aj
(4) (or (5)) is a type of eigenvalue problem for ω2.

12. We can rewrite all of this in matrix form
We can rewrite all of this in matrix form. Treating Vij and Tij as square matrices, aj as column vectors & ω2 as eigenvalues (λ  ω2):
(Vijaj - ω2Tijaj ) = (i = 1,2,3, … n) (4)
becomes: Va = λTa (6)
(6): This is a variation of the more common eigenvalue problem:
 The effect of matrix V acting on an eigenvector a is to produce a multiple (λ) of the matrix T acting on the eigenvector a.

13. Va = λTa (6) Can show (for proofs, see text pages 242-245):
1. The eigenvalues λ = ω2 are all real and positive.
(Comes from the real & symmetric properties of V & T)
2. The eigenvectors a are all orthogonal and real.
3. Matrix of eigenvectors  A
a. A is orthogonal (A-1 = Ã)
b. A diagonalizes both T and V
That is, performing a similarity transformation on T:
A-1TA = ÃTA = 1 (unit matrix)
That is, performing a similarity transformation on V:
A-1VA = ÃVA = Vd (diagonal matrix; eigenvalues on diagonal)
 If T is already diagonal in our choice of generalized coordinates, to find the eigenvalues, it is sufficient to diagonalize V!!

14. Lagrangian: L = (½)m[(x1)2 + (x2)2] - (½)Vijxixj
Simple example: Mass m with 2 degrees of freedom, generalized coords x1 & x2:
Lagrangian: L = (½)m[(x1)2 + (x2)2] - (½)Vijxixj
(summation convention)
Vij = constants
T is already diagonal. So, to find the eigenvalues, we need to only diagonalize V. In this case, λ = mω2
 (V - λ1)a = (1)
 Determinant: V11 - λ V12
V V22 - λ = 0
Eigenvalues (Like the simple “two-level” problem in QM!):
λ1 = m(ω1)2 = (½){V11 + V22 + [(V11 + V22)2 + 4V12V12]½}
λ2 = m(ω2)2 = (½){V11 + V22 - [(V11 + V22)2 + 4V12V12]½}

15. Allowed oscillation (vibration) frequencies:
λ1 = m(ω1)2 = (½){V11 + V22 + [(V11 + V22)2 + 4V12V12]½}
λ2 = m(ω2)2 = (½){V11 + V22 - [(V11 + V22)2 + 4V12V12]½}
To get the eigenvectors (oscillation amplitudes) corresponding to these frequencies, put λi into the original eigenvalue equation & solve for the matrix elements aij of the eigenvalue matrix (whose columns are the eigenvectors) : (V - λ1)a = 0
or: aij(Vij – λiδij) = 0
with normalization: (ai1)2 +(ai2)2 = 1

16. Allowed oscillation (vibration) frequencies:
λ1 = m(ω1)2 = (½){V11 + V22 + [(V11 + V22)2 + 4V12V12]½}
λ2 = m(ω2)2 = (½){V11 + V22 - [(V11 + V22)2 + 4V12V12]½}
General solution for the matrix of eigenvectors:
a11 a21
A = a12 a22
 a11 = V12[(V11 - λ1)2 + (V12)2]-½
a12 = (V11 - λ1)[(V11 - λ1)2 + (V12)2]-½
a21 = (V22 - λ2)[(V22 - λ2)2 + (V21)2]-½
a22 = V21[(V22 - λ2)2 + (V21)2]-½

17. Limiting Case 1: Small off diagonal Vij:
We have the exact, general solution. However, its useful & educational to look at 2 limiting cases.
Limiting Case 1: Small off diagonal Vij:
V11 > V22 > 0 & V12 = V21  0; V12 << (V11 - V22)
Define: δ  V12(V11 - V22)-1 << 1. Expand various square roots to lowest order in the small quantity δ:
 Eigenvalues: λ1 = m(ω1)2 = V11 + V12δ
λ2 = m(ω2)2 = V22 - V12δ
Eigenvectors: a11 a21
A = a12 a22
 a11 = a22 = 1 – (½)δ a21 = - a12 = δ - (½)δ3

18. Limiting Case 2: Large off diagonal Vij:
V12 = V21 > V22 > 0 & (V11 - V22) << V12
Define: ε  (⅛)(V11 - V22)(V12)-1 << 1. Expand various square roots to lowest order in ε:
 Eigenvalues:
λ1 = m(ω1)2 = (½)(V11 + V22) + V12 + (V11 - V22)ε
λ2 = m(ω2)2 = (½)(V11 + V22) - V12 - (V11 - V22)ε
Eigenvectors: a11 a21
A = a12 a22
 a11 = a22 = (2)-½(1 + 2ε)
a21 = - a12 = (2)-½ (1 - 2ε)

19. Fig. a: Eigenvalues
λi vs. δ  V12(V11 - V22)-1
for 0 < δ < 3
Includes both limiting
cases in this range:
(In QM: “Level
repulsion” as the off-
diagonal interaction
is turned on.)

20. Fig. b: Eigenvector
components aij vs.
δ  V12(V11 - V22)-1 for
0 < δ < 3
Includes both limiting
cases in this range:

21. All of the preceding analysis: Assumed the eigenvalues λi are non-degenerate (no 2 of are equal!).
If there is degeneracy, then, this procedure must be modified. In this case, we cannot uniquely determine the components of the eigenvectors by this procedure. Actually, the eigenvectors in that case are not unique, but can be appropriately chosen from an infinite possible set.
Follow the so-called “Gram-Schmidt” orthogonalization procedure of linear algebra. Discussed in detail on pages of the text.

22. All of this discussion is identical to the Ch
All of this discussion is identical to the Ch. 5 discussion of degenerate eigenvalues of the inertia tensor, and to the procedure for finding the principal axes of the inertia tensor.
Hence the title of the section: “Principal Axis Transformation”