## Landau-Lifshitz Integrals: Part I

I've been in school this semester and noticed some tough integrals found in Landau-Lifshitz Mechanics, subsection 14 on Integration of the Equations of Motion. I've managed to complete a couple of them and I thought I'd put them here for records. This is also some practice in LaTeX for me which I'll need eventually for sure. If you have any questions feel free to comment.

I'm looking to solve an integral of them form:

$I=\int\limits_a^b \frac{dx}{\sqrt{((b-x)(x-a))}}$

This is a special case where the limits are the roots of the radicand. If we make the transformation:

$u=\sqrt{b-x},$

$du=-\frac{1}{2}\frac{dx}{\sqrt{b-x}}$

We can now use this to replace one of the factors in the denominator:

$I=-2\int\limits_{u_1}^{u_2} \frac{du}{\sqrt{x-a}}$

$u_2$ and $u_1$ are found by the transformation equation, which we can also use to get rid of the $x$ in the denominator. I'll find them below, but leave them as $u_2$ and $u_1$ for simplicity.

Now since:

$u^2=b-x$

$x=b-u^2$

$x-a=b-u^2-a$

$\sqrt{x-a}=\sqrt{b-a-u^2}$

Substituting back into our integral:

$I=2\int\limits_{u_2}^{u_1} \frac{du}{\sqrt{(b-a)-u^2}}$

$I= \frac{2}{\sqrt{b-a}}\int\limits_{u_2}^{u_1} \frac{du}{\sqrt{1-(\frac{u}{\sqrt{b-a}})^2}}$

Now another transformation can be made, which most conveniently can be chosen as:

$y=\frac{u}{\sqrt{b-a}}$

$dy=\frac{du}{\sqrt{b-a}}$

Using this transformation on our integral we obtain:

$I=\frac{2}{\sqrt{b-a}}\int\limits_{y_1}^{y_2} \frac{\sqrt{b-a}}{\sqrt{1-y^2}} dy$

$I=2\int\limits_{y_1}^{y_2} \frac{1}{\sqrt{1-y^2}} dy$

This integral is a standard integral for $arcsin(y)$, which we can derive by making the transformation:

$y=\sin{\theta},$

$dy=\cos(\theta)d\theta$

Back into our integral we have:

$I=2\int\limits_{\theta_1}^{\theta_2} \frac{\cos(\theta)}{\sqrt{1-\sin^2{\theta}}} d\theta$

$I=2\int\limits_{\theta_1}^{\theta_2} d\theta$

Since $1-\sin^2{\theta}=\cos^2{\theta}$. This is now a trivial integral, which is just the difference of the limits $\theta_2$ and $\theta_1$. The integral becomes:

$I=2(\theta_2-\theta_1)$

The task now is to find the values $\theta_2$ and $\theta_1$ by transforming back to the original equation. The $u_i$ limits are given by:

$u_2=\sqrt{b-b}=0$

$u_1=\sqrt{b-a}$

The $y_i$ limits are therefore:

$y_2 =\frac{\sqrt{b-a}}{\sqrt{b-a}}=1$

$y_1=\frac{0}{\sqrt{b-a}}=0$

and finally:

$\theta_2 = \arcsin{1} = \frac{\pi}{2}$

$\theta_1=\arcsin{0}=0$

Which gives us the result:

$I=2(\frac{\pi}{2}-0)$

$I=\pi$

This holds for all $a,b\in\mathbb{R}$ such that the limits are roots corresponding to the precise shape of $I$ given at the start. It's kind of amazing that it always comes out to $\pi$, a result which has been in the back of my mind since I first did this integral about a month ago.

## Release qls-3.1.11

This is a simple quick fix which removes the option for installing using the old (apparently deprecated) mysql_connect() function. It also removes the class files pertaining to the old MySQL versions. This will make sure that everyone who uses the script uses the improved extension (MySQLi).

The minimum supported PHP version has now been lifted to PHP 5.5.0.

## Release qls-3.1.10

Hey everyone,

After 5 years I've decided to finally fix all the bugs in the script and release a new stable version. I went through every bug report in the Bugs section of the forum and made sure I took a look at each one. I found that most of the bugs were caused by poor coding; let's just say I've gotten much better in 5 years. You can find the new links on the downloads page:

If you're updating from 3.1.9 just choose the qls-3.1.10-changed_files_only version.

Here is an excerpt from the CHANGELOG:

[Fix] Logging in falsely will now produce an error message
[Fix] Editing a file will now work with zero length files
[Fix] Session->clear_old_sessions() error in API syntax (Thanks to user Monter)
[Fix] Session->clear_old_sessions() now being used in qls.class.php
[Fix] Session->validate_session() using wrong $information index (Thanks to user Monter) [Fix] User->validate_password() uses min_password now instead of min_username (Thanks to user dirmhirn) [Fix] auth_admin_edit_page was missing from run_form() in admin_add_mask_form.php (Thanks to user Mamoune) [Fix] Various installation issues including error reporting [Fix] Removed dual entries in the language file (Thanks to user Wormnest) [Feature] Added ACTIVATION_BODY_SUCCESS to English.lang.php [Feature] An email is now sent when an administrator activates a user account (Thanks to user Wormnest) [Fix] Change password bug fixed where a blank username still sent the form (Thanks to user jeffr and lulu) [Fix] Added activation time when activating users from admin panel (Thanks to user Wormnest) [Fix] Password matching failed with special characters on registration (Thanks to user antonio1) [Fix] Changed encodeURI to encodeURIComponent in admin_panel.php (Thanks to user BillG) [Info] Dropped support for updating from anything below 3.1.9 [Fix] Added ADMIN_EDIT_USER_NO_AUTH to English.lang.php [Fix] check_username_existance() was changed to check_username_existence() in User.class.php [Info] Indented everything that was not in my previous coding style [Info] Added variable declaration on line 281 in Security.class.php [Info] Added variable declaration on line 375 in qls.class.php [Info] Changed documentation for constructor in MySQL.class.php and MySQLie.class.php [Info] Dropped support for PostgreSQL (will return in 4.0.0) [Fix] Lines 75 and 116 of Group.class.php was not referencing the qls class properly [Fix] Increased the minimum version to PHP 5.2.0 [Fix] Website error in admin_panel.php [Fix] Random variable on line 707 named$save removed in User.class.php
[Fix] Replaced ereg_replace() with str_replace() in SQL.class.php line 59
[Fix] Replaced all email matching with the filter_var() function
[Fix] Variable name mismatch in Admin.class.php (various locations)
[Fix] Any reference to the .com version of the site removed finally!

This is an extensive list! There were no new real features added in this version, but I think I found most of the errors. Users who reported the bugs/provided a fix in the forum have been included in the list, thanks for your help!

## This Blog and Login Script Development

This Blog

This blog will be dedicated to my hobbies and projects! These include the Quadodo Login Script, and possibly some video game development I will work on (amateur). Also, there might be some useful information I find on the Internet posted here in tutorial format.

I might also occasionally discuss some advanced physics topics as I learn about them.

So there has really been no development in a few years on the Quadodo Login Script project. I intend to change this very soon, after I get some free time. I'm currently getting a degree in Physics which is consuming my time. This project was always a hobby project and since there is a stable version it isn't on the top of my list of things to complete.

That being said it is still important to me, and I want to fix some of the issues in the code (just bad coding in general) and in the design of the script. I want to be able to implement a better API as well as better support for mods and extra languages. Currently the script doesn't have great support for modifications, and the update structure is quite awful. I want to build something that can last a few more years without input.

I've gone over all of the suggestions in the UserVoice page and have considered most of them.

phpBB Integration

Integration with phpBB would be an important feature. Since phpBB has its own user management I'm feeling as if the administration control panel would be useless. I think I will integrate it and then disable the administration control panel if the option is selected at install. This would allow them to manage users via phpBB, but still control which pages are protected.

As I'm writing this I'm actually thinking the control panel might be useful though, if people wanted to add pages or send invitations still. I'll decide on something eventually.

Payment Gateway

This has been an important topic for me as well. Years ago I couldn't really understand how to integrate payments into the system effectively. Now that I have a lot more experience and problem solving ability (thanks Physics) I'm thinking that PayPal would be the first thing integrated. Other suggestions would be appreciated and maybe I can integrate them as well.

Anyway, there will be more to come after exams and during the summer.