2024-04-09 Coupled Oscillations Cont.

Summary from 2024-04-05 General Coupled Oscillations

• Normal frequencies given by ${\omega }_r$ (eigenfrequencies)
• Eigenvectors ${\vec{a}}_r$ with $a_{gr}$ composition

$$\sum _j(A_{jk}{q}_{j}j+m_{jk}{\ddot{q}}_j)=0$$

Guessing

$$q_j(t)=a_j{e}^{i(\omega t-\delta )}$$

Given this

$$\sum _j(A_{jk}-{\omega }^2{m}_{jk})a_j=0$$ $$\det |A_{jk}-{\omega }^2{m}_{jk}|=0$$

to find ${\omega }_r$

Revisiting first example of coupled oscillations with new notation

figure 1

$$T=\frac{1}{2}m{\dot{x}}_1^2+\frac{1}{2}m{\dot{x}}_2^2$$ $$u=\frac{1}{2}\kappa x_1^2+\frac{1}{2}\kappa x_2^2+\frac{1}{2}{\kappa }_{12}(x_1-x_2{)}^2$$ $$u=\frac{1}{2}(\kappa +{\kappa }_{12})x_1^2+\frac{1}{2}(\kappa +{\kappa }_{12})x_2^2-{\kappa }_{12}{x}_1{x}_2$$ $$\begin{array}{cc}{q}_1=x_1& {\dot{q}}_1={\dot{x}}_1\\ q_2=x_2& {\dot{q}}_2={\dot{x}}_2\end{array}$$ $$T=\frac{1}{2}m({\dot{x}}_1^2+{\dot{x}}_2^2)$$ $$=\frac{1}{2}m({\dot{q}}_1^2+{\dot{q}}_2^2)$$ $$T=\frac{1}{2}\sum _{jk}{m}_{jk}{\dot{q}}_j{\dot{q}}_k$$ $$\frac{1}{2}{m}_{11}{\dot{q}}_1{\dot{q}}_1\leftrightarrow \frac{1}{2}m{\dot{q}}_1^2$$ $$m_{11}=m=m_{22}$$ $$m_{12}=m_{21}=0$$

Revisit this

$$u=\frac{1}{2}(\kappa +{\kappa }_{12})x_1^2+\frac{1}{2}(\kappa +{\kappa }_{12})x_2^2-{\kappa }_{12}{x}_1{x}_2$$ $$A_{jk}=\frac{{\partial }^{2}u}{\partial q_j\partial q_k}$$ $$A_{11}=\kappa +{\kappa }_{12}$$ $$A_{22}=\kappa +{\kappa }_{12}$$ $$A_{12}=A_{21}=-{\kappa }_{12}$$ $$\det |A_{jk}-{\omega }^2{m}_{jk}|=0$$ $$\det \left(\begin{array}{cc}(\kappa +{\kappa }_{12})-{\omega }^{2}m& {\kappa }_{12}\\ -{\kappa }_{12}& (\kappa +{\kappa }_{12})-{\omega }^{2}m\end{array}\right)=0$$ $$\omega =\sqrt{\frac{\kappa +{\kappa }_{12}\pm {\kappa }_{12}}{2}}$$ $${\omega }_1=\sqrt{\frac{\kappa +2{\kappa }_{12}}{m}}$$

and

$${\omega }_2=\sqrt{\frac{\kappa }{m}}$$

The eigenvectors ${\vec{a}}_r$ will form an orthogonal set. This set can be normalized to create an orthonormal set. This is shown mathematically in section 12. 5 (pg. 481) of the textbook. In this process to create an orthonormal set of eigenvectors, the ambiguity of $q_j$ is removed by the normalization of $a_{jr}$, so we must introduce a scale factor, ${\alpha }_r$which is dependent on the initial conditions.

so

$$q_j(t)\sum _r{a}_{jr}\cos ({\omega }_{r}t-{\delta }_r)$$

or-

$$q_j(t)=\sum _r{a}_{jr}{e}^{i(\kappa rt-\delta r)}$$

Normalized:

$$q_j(t)=\sum _r{\alpha }_r{a}_{jr}{e}^{i({\omega }_{r}t-{\delta }_r)}$$ $$=\sum _r{\beta }_r{a}_{jr}{e}^{i{\omega }_{r}t}$$

Where ${\beta }_r$ is a new scale factor (complex) which incorporates phase ${\delta }_r$

• Defining a quantity ${\eta }_r(t)={\beta }_r{ϵ}^{i{\omega }_{r}t}$ such that

$$q_j(t)=\sum _r{a}_{jr}{\eta }_r(t)$$

• This recovers our normal coordinates (only oscillates at one frequency)
• ${\eta }_r(t)$ is normal coordinate

From our example

$$\det \left[\begin{array}{cc}\kappa +{\kappa }_{12}-m{\omega }^2& -{\kappa }_{12}\\ -{\kappa }_{12}& \kappa +{\kappa }_{12}-m{\omega }^2\end{array}\right]=0$$

Found ${\omega }_1$ and ${\omega }_2$

To find $a_{jr}$ components we only need 1 equation (for this 2 mass case) since we are only after a ratio of components $\frac{a_{1r}}{a_{2r}}$

$$r=1$$ $${\omega }_1^2={\left(\sqrt{\frac{\kappa +2{\kappa }_{12}}{m}}\right)}^2=\frac{\kappa +2{\kappa }_{12}}{m}$$

Eigenvector ${\vec{a}}_1$ with components $a_{11}$ and $a_{21}$ Choose first equation:

$$(A_{11}-m_{11}{\omega }_1^2)a_{11}+(A_{12}-m_{12}{\omega }_1^2)a_{12}=0$$ $$[\kappa +{\kappa }_{12}-m(\kappa +2{\kappa }_{12})]a_{11}-{\kappa }_{12}{a}_{21}=0$$ $$-{\kappa }_{12}{a}_{11}-{\kappa }_{12}{a}_{21}=0$$ $$-a_{11}=a_{21}$$

The general equation of motion:

$$q_j(t)=\sum _r{a}_{jr}{\eta }_r(t)$$ $$q_1(t)=a_{11}{\eta }_1+a_{12}{\eta }_2$$ $$q_2(t)=a_{21}{\eta }_1+a_{22}{\eta }_2$$

Use the fact that $-a_{11}=a_{21}$

Rewrite:

$$q_1(t)=a_{11}\eta 1+a_{22}{\eta }_2$$ $$q_2(t)=-a_{11}{\eta }_1+a_{22}{\eta }_2$$

So now we can find normal coordinates;

$$q_1+q_2=2a_{22}{\eta }_2$$ $$q_1-q_2=2a_{11}\eta 1$$ $${\eta }_2=\frac{1}{2a_{22}}(q_1+q_2)$$ $${\eta }_1=\frac{1}{2a_{11}}(q_1-q_2)$$

Can use these relationships to ask what conditions are needed for only 1 coordinate to oscillate. ${\eta }_1=0$ if $q_1=q_2$, therefore $q_1=q_2$ makes only ${\eta }_2$ oscillate with frequency ${\omega }_2=\sqrt{\frac{\kappa }{m}}$, which is “in-phase” oscillation. ${\eta }_2=0$ if $q_1=-q_2$ therefore, opposite to the above, $q_1=-q_2$ makes only ${\eta }_1$ oscillate with frequency ${\omega }_1=\sqrt{\frac{\kappa +{\kappa }_{12}}{m}}$, an “out-of-phase” oscillation.

Molecular Vibrations (12.7)

Good application of small oscillations In 3 dimensions a molecule with n atoms will have 3n degrees of freedom.

3 degrees of freedom for translation, 3 degrees of freedom for rotation

3n-6 degrees of vibrational freedom

If the molecule is colinear then there is 3n-5 degrees of vibration freedom.