# CM Course Notes

Here is a conglomerate of notes gathered for the graduate CM class.

## 220 Notes

Lagrangian Mechanics

Method for finding normal modes of system

Since we expect oscillatory motion of a normal mode (where ω is the same for both masses), we try: $x_1(t) = A_1 e^{i \omega t} \\ x_2(t) = A_2 e^{i \omega t}$

Euler-Lagrange Equation of system without constraints:

$\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q}_j} \right ) = \frac {\partial L}{\partial q_j}$

Origin of Canonical Momentum:

$p = \left( p - \frac{e}{c} A \right), \text{ where p inside the brackets is the quantum mechanical momentum } p = -i \hbar \nabla \text{ and A is the gauge invariant vector potential.}$ 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

## Gallery

Lecture Notes

Classical Mechanics by Prof. Eric D'Hoker.