Can i append an already existing div to another already existing div?

Check this jsFiddle for a quick POC. Apparently it does.

The trick is to pass the object reference, not just the object id, like so:

$('#menu_form').on('click', function(){
    $('#form_div').append($('#display_div'));
});

You could also pass the current object, using this:

$('#menu_form').on('click', function(){
    $('#form_div').append(this);
});

Using .appendTo()

$('#menu_form').on('click', function(){
   $('#form_div').appendTo('#display_div');  // appendTo -> selector
});

Using .append()

$('#menu_form').on('click', function(){
   $('#display_div').append( $('#form_div') ); // append -> object
});

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