Small Coupled Oscillations

1. Small Coupled Oscillations

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

3. 6.1 • 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 ηi

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

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

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

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

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

9. 6.2 6.3 • 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

10. 6.2 6.3 • 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:

11. 6.2 6.3 • 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.

12. 6.4 • Example: longitudinal oscillations of a CO2 molecule • CO2 is a linear molecule; we will model it as follows: • The Lagrangian

13. 6.4 • Example: longitudinal oscillations of a CO2 molecule • Secular equation:

14. 6.4 • Example: longitudinal oscillations of a CO2 molecule • Eigen-vectors:

15. 6.4 • Example: longitudinal oscillations of a CO2 molecule • Eigen-vectors:

16. 6.4 • Example: longitudinal oscillations of a CO2 molecule • Normal coordinates:

17. 6.5 • 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:

18. 6.5 • 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

19. Questions?

20. 6.2 6.3 Normal coordinates

21. 6.1 • 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

22. 6.1 • 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