1. THIS LECTURE single From single to coupled oscillators

2. Coupled pendula

3. Propagation of energy in space

4. Two thin rods connected by a pivot “Chaotic” motion Coupled pendula

5. Coupled oscillators Normal modes Modes of vibration in which each oscillator vibrates with same frequency Coupled motion implies energy exchange

6. Two coupled oscillators Newton equations k’ k k x1x1 x2x2 A 1, A 2,  1 and  2 are constants that depend on initial conditions

7. Two coupled oscillators Newton equations k’ k k x1x1 x2x2 A 1, A 2,  1 and  2 are constants that depend on initial conditions

8. Two coupled oscillators Newton equations k’ k k x1x1 x2x2 A 1, A 2,  1 and  2 are constants that depend on initial conditions

9. NORMAL MODES Each mass vibrates with the same frequency NORMAL MODE 1 Masses are in phase and vibrate with frequency Masses are in antiphase and vibrate with frequency NORMAL MODE 2

10. Problem At t =0, one mass is moved a distance 2a, but the second mass is held fixed. Find the time-dependence of x 1 and x 2. T2T2 T1T1

11. Two coupled oscillators: general case Newton equations k2k2 k1k1 k3k3 x1x1 x2x2 M m To find the normal modes, we assume that x 1 and x 2 are harmonic functions with same frequency . Normal modes

12. Two coupled oscillators: general case Newton equations k2k2 k1k1 k3k3 x1x1 x2x2 M m To find the normal modes, we assume that x 1 and x 2 are harmonic functions with same frequency . Normal modes Newton equations

13. Two coupled oscillators: general case Newton equations k2k2 k1k1 k3k3 x1x1 x2x2 M m This system of equations has solution if the determinant of its coefficients vanishes This is a quadratic equation in the variable . It has two solutions,  1 and  2, corresponding to two normal modes.

14.  1 and  2 corresponding to two normal modes NORMAL MODE 1NORMAL MODE 2 The most general solution is the superposition of the two modes. Two coupled oscillators: general case k2k2 k1k1 k3k3 x1x1 x2x2 M m

15. Crystals Coupled oscillators

16. N coupled oscillators z y x Method to describe the motion Consider separate motion along x, y, z Write down Newton equations for the displacement of each mass along a given direction, for example x. Displacements are called x 1, x 2, x 3 …x N. Normal modes Determine the normal modes for each variable x 1, x 2 …x N to obtain equations that describe harmonic motion