1. Small Coupled Oscillations

2. Types of motion Each multi-particle body has different types of degrees of freedom: translational, rotational and oscillatory

3. Formulation of the problem Let us consider a many-particle Lagrangian The system is stable, if each particle has a stable equilibrium position We assume small deviations from equilibrium η i 6.1

4. Formulation of the problem Kinetic energy of the system Potential energy of the system 6.1

5. Formulation of the problem We assume that this term does not vanish The Lagrangian of the system Equations of motion 6.1

6. Normal coordinates We have a system of linear ordinary differential equations of the 2 nd order A natural choice of a trial solution Equations of motion result in 6.2 6.3

7. Normal coordinates Let us consider diagonal terms l = k 6.2 6.3

8. Normal coordinates If real α and β are assumed to be some velocities, then this expression has a form of a kinetic energy, which is always positively defined Thus if 6.2 6.3

9. Normal coordinates Equations of motion do not have exponentially growing solutions This can be true only for two diagonal matrices We have a freedom of normalization for matrix a ; let us impose the following normalization: Recall Then 6.2 6.3

10. Normal coordinates Equations of motion : We completely diagonalized our problem We have a generalized eigen-value problem Eigen-values of the problem are solutions of the secular equation: Eigen-vectors: 6.2 6.3

11. Normal coordinates Secular equation As the number of generalized coordinates increases, the power of the secular equation grows For very large systems, there are two ways to calculate eigen-values: analytical application of the group theory and computer calculations Modern applications: molecular vibrational spectroscopy, solid-state vibrational spectroscopy, etc. 6.2 6.3

12. Example: longitudinal oscillations of a CO 2 molecule CO 2 is a linear molecule; we will model it as follows: The Lagrangian 6.4

13. Example: longitudinal oscillations of a CO 2 molecule Secular equation: 6.4

14. Example: longitudinal oscillations of a CO 2 molecule Eigen-vectors: 6.4

15. Example: longitudinal oscillations of a CO 2 molecule Eigen-vectors: 6.4

16. Example: longitudinal oscillations of a CO 2 molecule Normal coordinates: 6.4

17. Forced oscillations For open systems, we introduce generalized forces For each generalized coordinate, there is a component of a force We can introduce modified generalized forces for each normal coordinate Total work done Equations of motion: 6.5

18. Forced oscillations Let us consider a periodic external force We look for a solution in the following form: After substitution into the equation of motion For generalized coordinates Resonance 6.5

19. Questions?

20. Normal coordinates 6.2 6.3

21. The independent coordinates of a rigid body Let us consider a many-particle Lagrangian The system is stable, if each particle has a stable equilibrium position We assume small deviations from equilibrium 6.1

22. The independent coordinates of a rigid body Let us consider a many-particle Lagrangian The system is stable, if each particle has a stable equilibrium position 6.1