Despite the fact that wordpress uses PHP, I must develop a web application inside a wordpress site I curently have, to ensure that I do not have to redo the website on your own, and may make use of the same elements of design I have already developed.

It is possible to method to have certain pages on my small existing wordpress site run Grails without anyone's knowledge, after which goes the right HTML elements to ensure that it may make use of the existing buttons, divs, etc.?

Grails and PHP it totatlly different techologies. Grails is really a framework designed in and uses Groovy/Java, requires different web-server, etc.

Btw, there's one other way: you are able to hide this two applications behind one frontend (nginx, or something like that), and map (proxy) different web addresses to various applications, some web addresses to wordpress, some to grails application.