If you change your single quote marks in to double quote marks it should work :
$cat_id = get_cat_id('library');
wp_dropdown_categories("hierarchical=1&parent=$cat_id");
but if you really want to make it more flexible you can phrase your arguments as an array:
$args = array(
'hierarchical' => 1,
'parent' => get_cat_id('library'));
wp_dropdown_categories($args);
and if you want to make it even more flexible to get the current category’s children you can use get_query_var('cat');
assuming that you are in your category.php file, so:
$args = array(
'hierarchical' => 1,
'parent' => get_query_var('cat'));
wp_dropdown_categories($args);