Difference between revisions of "CM Course Notes"
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+ | Origin of Canonical Momentum:<div class="mw-collapsible mw-collapsed" style="width:750px"> | ||
+ | \[p = \left( p - \frac{e}{c} A \right), \text{ where p inside the brackets is the quantum mechanical momentum } p = -i \hbar \nabla \text{ and A is the gauge invariant vector potential.}\] | ||
+ | Canonical momentum is not the momentum fired out of a cannon, but rather the momentum that satisfies $p = \frac{\partial \mathcal{L}}{ \partial \dot{q}}$, where $\mathcal{L}$ is the Lagrangian of the system. | ||
+ | :<math>H=\frac{1}{2m} \left(p-\frac{qA}{c}\right)^2 +q\phi</math> | ||
+ | </div> | ||
'''Hamiltonian Mechanics''' | '''Hamiltonian Mechanics''' | ||
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'''Lecture Notes''' | '''Lecture Notes''' | ||
− | [http://www.pa.ucla.edu/sites/default/files/files/dhoker%20lecture%20notes/ | + | [http://www.pa.ucla.edu/sites/default/files/files/dhoker%20lecture%20notes/mechanics2015.pdf Classical Mechanics] by Prof. Eric D'Hoker<ref>http://www.pa.ucla.edu/content/eric-dhoker-lecture-notes, retrieved 10th November 2015. </ref>. |
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Latest revision as of 07:35, 11 November 2015
Here is a conglomerate of notes gathered for the graduate CM class.
Contents |
[edit] 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: \[ x_1(t) = A_1 e^{i \omega t} \\ x_2(t) = A_2 e^{i \omega t} \]
\[\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q}_j} \right ) = \frac {\partial L}{\partial q_j} \]
\[p = \left( p - \frac{e}{c} A \right), \text{ where p inside the brackets is the quantum mechanical momentum } p = -i \hbar \nabla \text{ and A is the gauge invariant vector potential.}\] Canonical momentum is not the momentum fired out of a cannon, but rather the momentum that satisfies $p = \frac{\partial \mathcal{L}}{ \partial \dot{q}}$, where $\mathcal{L}$ is the Lagrangian of the system. \[H=\frac{1}{2m} \left(p-\frac{qA}{c}\right)^2 +q\phi\]
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
[edit] Exercises
[edit] General Identities
[edit] Gallery
Lecture Notes
Classical Mechanics by Prof. Eric D'Hoker[1].
[edit] References
- ↑ http://www.pa.ucla.edu/content/eric-dhoker-lecture-notes, retrieved 10th November 2015.