Difference between revisions of "CM Course Notes"
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==220 Notes== | ==220 Notes== | ||
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| + | '''Lagrangian Mechanics''' | ||
| + | Method for finding normal modes of system <div class="mw-collapsible mw-collapsed" style="width:750px"> | ||
| + | |||
| + | Since we expect oscillatory motion of a normal mode (where ω is the same for both masses), we try: | ||
| + | \[ | ||
| + | x_1(t) = A_1 e^{i \omega t} \\ | ||
| + | |||
| + | x_2(t) = A_2 e^{i \omega t} | ||
| + | \] | ||
| + | </div> | ||
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| + | Euler-Lagrange Equation of system without constraints: <div class="mw-collapsible mw-collapsed" style="width:750px"> | ||
| + | :<math>\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q}_j} \right ) = \frac {\partial L}{\partial q_j} </math> | ||
| + | |||
| + | </div> | ||
| + | |||
| + | '''Hamiltonian Mechanics''' | ||
| + | |||
| + | Hamiltonian i.t.o. Lagrangian: | ||
| + | <div class="mw-collapsible mw-collapsed" style="width:750px"> | ||
| + | :<math>H = \dot{\mathbf{q}} \frac{\partial \mathcal{L}}{\partial \dot{\mathbf{q}}} - \mathcal{L} </math> | ||
| + | </div> | ||
| + | |||
| + | Transform from rotating to fixed frame: | ||
| + | <div class="mw-collapsible mw-collapsed" style="width:750px"> | ||
| + | \[ | ||
| + | \vec{v}' = \vec{v} - \vec{\omega} \times \vec{r} | ||
| + | \] | ||
| + | </div> | ||
| + | |||
| + | '''Relativity''' | ||
| + | Write the product $\gamma\beta$ in terms of only $\gamma$:<div class="mw-collapsible mw-collapsed" style="width:750px"> | ||
| + | :<math>\gamma \beta = \sqrt{\frac{\beta^2}{1- \beta^2}}= \sqrt{\frac{A}{1-\beta^2} + B} \\ | ||
| + | \text{In the words of the notorious J.D. Jackson, ''we see'' that }A = 1 \text{ and }B = -1 \text{ , such that } \\ \gamma \beta = \sqrt{\frac{1}{1-\beta^2} - 1}=\sqrt{\gamma^2 - 1} </math> | ||
| + | E.g.: Lim#3021 | ||
| + | </div> | ||
| + | <br/> | ||
===''<small>Exercises</small>''=== | ===''<small>Exercises</small>''=== | ||
Revision as of 23:19, 14 April 2014
Here is a conglomerate of notes gathered for the graduate CM class.
Contents |
220 Notes
Lagrangian Mechanics
Method for finding normal modes of systemSince we expect oscillatory motion of a normal mode (where ω is the same for both masses), we try: \[ x_1(t) = A_1 e^{i \omega t} \\ x_2(t) = A_2 e^{i \omega t} \]
\[\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q}_j} \right ) = \frac {\partial L}{\partial q_j} \]
Hamiltonian Mechanics
Hamiltonian i.t.o. Lagrangian:
\[H = \dot{\mathbf{q}} \frac{\partial \mathcal{L}}{\partial \dot{\mathbf{q}}} - \mathcal{L} \]
Transform from rotating to fixed frame:
\[ \vec{v}' = \vec{v} - \vec{\omega} \times \vec{r} \]
Relativity
Write the product $\gamma\beta$ in terms of only $\gamma$:\[\gamma \beta = \sqrt{\frac{\beta^2}{1- \beta^2}}= \sqrt{\frac{A}{1-\beta^2} + B} \\ \text{In the words of the notorious J.D. Jackson, ''we see'' that }A = 1 \text{ and }B = -1 \text{ , such that } \\ \gamma \beta = \sqrt{\frac{1}{1-\beta^2} - 1}=\sqrt{\gamma^2 - 1} \] E.g.: Lim#3021
Exercises
General Identities
