I'm assuming that you are drawing a 3D scene and want to change it's orientation, rather than having a 2D image which you wish to rotate.
The short answer is that it takes place in two parts. You need to store an orientation of your scene as a 4x4 matrix (homogeneous matrix - search for it if you don't know what that is). You first need to write code that translates a mouse drag into a change of that 4x4 matrix. So when the mouse is dragged up apply an appropriate rotation or whatever to the matrix.
Then you need to redraw the scene, but using the new transformed 4x4 matrix. Use glMatrixMode to specify which matrix (use either GL_PROJECTION or GL_MODELVIEW) and then functions like glMultMatrixf() to manipulate the appropriate matrix.
If that didn't make sense pick up an OpenGL tutorial on how to rotate scenes. OpenGL and JOGL are close enough that methods from OpenGL work in JOGL.