1. Physics 319 Classical Mechanics
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 21
G. A. Krafft
Jefferson Lab

2. Lagrangian Small Oscillation Theory
Method for solving problems where several coupled oscillations present
Steps are
Write Lagrangian for several oscillations including coupling. If needed go into small oscillation limit
Solve for system oscillation “normal mode” frequencies
Solve for oscillation amplitude vector for each normal mode
Go into coordinates, the so-called normal mode coordinates, where the oscillations de-couple, to solve initial conditions and time dependences

3. Two Masses and Three Springs
Forces and equation of motion are
Introduce 2 component “vector” describing state of system

4. Equations of Motion in Vector Form
Equations of motion are

5. Sinusoidal Ansatz
As we have done many times before assume sinusoidal solutions of general form
Simultaneous Linear Equations! Solution method from Linear Algebra
Gives possible “normal mode” oscillation frequencies. Then solve for associated (eigen)vector.

6. Case of Identical Masses and Springs
Normal mode frequency problem an eigenvalue problem. Solve normal mode (also called secular) equation

7. First Normal Mode Take minus sign solution
Back in original matrix equation
Such an oscillation in the system is the symmetric mode
Masses move in the same direction with the middle spring unextended. Oscillation frequency “obviously” satisfies

8. Second Normal Mode Take plus sign solution
Now normal mode eigenvector is
Such an oscillation in the system is the antisymmetric mode
Masses move in the opposite directions with the middle spring extended twice as much as the other two.

9. In Pictures

10. General Solution
General solution for motion determined by 4 initial conditions, giving the real and imaginary parts of A and B
Picture of general motion

11. Normal Mode Coordinates
General motion is simplified if go into coordinates tied to the normal mode eigenvector pattern. Define
These combinations will only oscillate at the normal mode frequencies ω± separately, ξ1 at ω− and ξ2 at ω+
By going into the normal mode coordinates, the coupled oscillations problem becomes decoupled!

12. Case of Weak Coupling Expect slight frequency shifts in oscillators
Normal mode eigenvectors are the same symmetric and antisymmetric combinations that we saw before.

13. General Solution Place following boundary conditions on solution
Then get
Phase delayed oscillations with amplitude that goes from one degree of freedom to the other and back again