Initiate class only once as global and add functions to class from outside

Don't know better title to describe my problem.

So here's the long version:

Problem 1:

I have a php class with all the stuff in it to control a page (Languagemanagement, functions, Databasecoection etc.)

And im loading content with ajax.
Right now i have to initiate the same class for each ajax file. Which seems stupid to me.

Isn't there a way to initiate the class once as global and use it in ajax loaded files too?

Something like:
(Mainpage)
$class=new myClass();

and in Ajaxfile

$class->dosomething();

$GLOBALS not working and a $_SESSION can't store a class.

Problem 2:

How can i add new functions to my class without touching the classfile?

Example:

How do i need to create addFunction to get the needed result?

Thanks in advance