1. Physics 319 Classical Mechanics
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 22
G. A. Krafft
Jefferson Lab

2. 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

3. Beats
In normal mode view have oscillators at slightly different frequencies. Beating is constructive and destructive interference

4. Lagrangian Version of Two Carts
Euler-Lagrange equations of motion are identical to before

5. Double Pendulum Oscillations
Double Pendulum Lagrangian
Small Oscillation (Linearization) Approximation
(0,0) l1 l2

6. Equations of Motion
The Euler-Lagrange equations of motion are written in the matrix form as
Same solution method as before

7. Equal Mass and Length Case
When m1 = m2 and l1 = l2

8. Normal Modes First Normal Mode Symmetric Displacements
Note: amplitudes different in the normal mode!

9. Symmetric Displacement

10. Second Normal Mode
Other normal mode
Antisymmetric normal mode

11. Antisymmetric Displacement

12. n Coupled Bodies
Use Lagrangian description to analyze general case of n coupled bodies. The general displacement is described by a column vector
Evaluate the kinetic energy
Defines the M matrix as before

13. General Potential Matrix
For small oscillations about an equilibrium point, the dominant contribution for the potential follows from a Taylor expansion of the potential energy function
Euler-Lagrange equations yield, for each index i
This is the familiar matrix equation

14. Solution Method
Find n normal mode frequencies by solving the secular or characteristic equation
Find n normal mode vectors
Write general solution as the sum over the normal modes. Expansion coefficients determined by the initial conditions

15. Three Coupled Oscillators
Solve the problem of three coupled oscillators
Kinetic energy easy
Gravitation potential easy

16. Equations of Motion
Spring potential energy
Dynamical matrices

17. Secular Equation
First normal mode totally symmetric

18. Second normal mode, antisymmetric like
Third normal mode, also antisymmetric like

19. From Taylor

20. Resolution into Normal Modes
Given the time displacement of the total system and that you know the normal mode eigenvectors, one can easily determine the normal mode amplitude and phase. Simply perform the sum using the eigenvector of the mode