Sunday, January 22, 2012


Let's follow up on a few previous posts (a calculation of d/dt of the total energy in a gravitational field, a post about typesetting math, and an introductory exploration of LaTeX). I'm grateful for a comment from a reader, with a link to this page, which makes it pretty clear that, yes, you can use their server to get the script in your pages.

So that's what we'll do. I had to modify the LaTeX commands a bit for this but it's mostly the same as before.

[ UPDATE: I do see a problem, now. The script applies to all LaTeX on the page, which if you go to the main page for the blog, includes the previous post... Just click on the post title, to see the original formatting code. ]

$\mathbf{F} = m\mathbf{a}= m\ddot{\mathbf{r}}
= m\frac{d^2}{dt^2}\mathbf{r}$

$\mathbf{F} = -$$\nabla$$V(\mathbf{r})$

$E = \frac{1}{2}m|\dot{\mathbf{r}}|^2+ V$

$\frac{d}{dt} E = \frac{d}{dt} ( \frac{1}{2}m
|\dot{\mathbf{r}}|^2 + V )$ $= ?$

$|\dot{\mathbf{r}}|^2 = |\dot{\mathbf{r}}|
|\dot{\mathbf{r}}| = \dot{\mathbf{r}}

= \frac{1}{2}m\frac{d}{dt}(\dot{\mathbf{r}}
\cdot\dot{\mathbf{r}})$$ = m\dot{\mathbf{r}}\cdot\ddot{\mathbf{r}}$$ = \dot{\mathbf{r}}\cdot-($$\nabla$$V$$)$

$\nabla$$V$ $= < \frac{\partial{V}} {\partial{x}}, \frac{\partial{V}} {\partial{y}},\frac{\partial{V}} {\partial{z}} >$

$\dot{\mathbf{r}}$ = $<$ $\frac{dx}{dt}$, $\frac{dy}{dt}$,$\frac{dz}{dt}$$ >$

$\nabla$$V$ $\cdot$$\dot{\mathbf{r}}$ $= < \frac{\partial{V}} {\partial{x}} \frac{dx}{dt}, \frac{\partial{V}} {\partial{y}} \frac{dy}{dt}, \frac{\partial{V}} {\partial{z}}\frac{dz}{dt}>$$=\frac{d}{dt}V$

$\frac{d}{dt} E = \frac{d}{dt} ( \frac{1}{2}m
|\dot{\mathbf{r}}|^2 + V )$ $= ?$

$\frac{d}{dt}$E $= \dot{\mathbf{r}}$$\cdot$(-$\nabla$V) + $\nabla$V $\cdot$ $\dot{\mathbf{r}}$ = 0

No comments: