1. Hour 33 Coupled Oscillators I
Physics 321
Hour 33
Coupled Oscillators I

2. Example I x1 x2 k1 k2 k3
𝑇= 1 2 𝑚 1 𝑥 𝑚 2 𝑥 2 2
𝑈= 1 2 𝑘 1 𝑥 𝑘 2 𝑥 2 − 𝑥 𝑘 3 𝑥 2 2
𝐿= 1 2 𝑚 1 𝑥 𝑚 2 𝑥 2 2 − 1 2 𝑘 1 𝑥 1 2
− 1 2 𝑘 2 𝑥 2 − 𝑥 − 1 2 𝑘 3 𝑥 2 2

3. Example I x1 x2 k1 k2 k3
𝐿= 1 2 𝑚 1 𝑥 𝑚 2 𝑥 2 2 − 1 2 𝑘 1 𝑥 1 2
− 1 2 𝑘 2 𝑥 2 − 𝑥 − 1 2 𝑘 3 𝑥 2 2
𝑚 1 𝑥 1 =− 𝑘 1 𝑥 1 + 𝑘 2 𝑥 2 − 𝑥 1 =− 𝑘 1 + 𝑘 2 𝑥 1 + 𝑘 2 𝑥 2
𝑚 2 𝑥 2 =− 𝑘 3 𝑥 2 − 𝑘 2 𝑥 2 − 𝑥 1 =− 𝑘 2 + 𝑘 3 𝑥 2 + 𝑘 2 𝑥 1

4. Example I x1 x2 k1 k2 k3
𝑚 𝑚 𝑥 𝑥 2 =− 𝑘 1 + 𝑘 2 − 𝑘 2 −𝑘 2 𝑘 2 + 𝑘 𝑥 1 𝑥 2
𝐌 𝑥 =−𝐊 𝑥

5. Normal Modes Assume solutions of the form: 𝑥 1 𝑡 = 𝐴 1 sin 𝜔𝑡+ 𝜑 1
Note that 𝜔 is the same in both equations! – This defines a “normal mode.”
Then 𝐌 𝑥 =− 𝜔 2 𝐌 𝑥 =−𝐊 𝑥

6. Normal Modes - Method 1 𝐌 𝑥 =− 𝜔 2 𝐌 𝑥 =−𝐊 𝑥 → det 𝐊− 𝜔 2 𝐌 =0
𝐌 𝑥 =− 𝜔 2 𝐌 𝑥 =−𝐊 𝑥
→ det 𝐊− 𝜔 2 𝐌 =0
𝑘 1 + 𝑘 2 − 𝜔 2 𝑚 1 − 𝑘 2 −𝑘 2 𝑘 2 + 𝑘 3 − 𝜔 2 𝑚 2 =0
Modified eigenvalue problem
Solve for 𝜔 2
Use 𝐊− 𝜔 2 𝐌 𝑥 1 𝑥 2 =0 to solve for 𝑥 1 and 𝑥 2
Easiest by hand.

7. Normal Modes - Method 2 𝐌 𝑥 =− 𝜔 2 𝐌 𝑥 =−𝐊 𝑥 𝐌 −1 𝐊 𝑥 = 𝜔 2 𝑥
𝐌 𝑥 =− 𝜔 2 𝐌 𝑥 =−𝐊 𝑥
𝐌 −1 𝐊 𝑥 = 𝜔 2 𝑥
→ det 𝐌 −1 𝐊− 𝜔 2 𝟏 =0
Standard eigenvalue problem
Eigenvalues are 𝜔 2
Eigenvectors are normal modes
Easiest with Mathematica

8. Examples
coupled oscillators.nb
normal coords.nb