It is said that in a limited war one of the main goals is to win the will of the people--to have them support you and come on board ideologically. How would one measure this kind of goal and know whether it was being achieved?
The phrase "limited war" has become a little misunderstood, at least with regard to the notion of "winning the hearts and minds" of the people in whose country the conflict is being fought. "Limited war" need not have anything to do with "hearts and minds." Rather, it refers to the scope of conflict in which the United States or some other country has chosen to become engaged. The last unlimited war in which the United States was involved was World War II, when the total defeat of Germany and Japan was the goal of the United States and its allies, and considerations of foreign fatalities was not a concern. All subsequent conflicts have been limited in terms of how far the United States has been willing to go to achieve some kind of victory.
A classic case of limited war was the first Persian Gulf War (Operation Desert Storm), in which the U.S. goal was not the total defeat of Iraq, but rather the liberation of Kuwait, which had been invaded by the Iraqi Army. Then-President George H.W. Bush made a point of ensuring the world that the United States goal was not to replace the government of Iraq by force, but rather to expel Iraqi forces from Kuwait while weakening its military. "Hearts and minds" played very little role in that limited war.
The more recent war in Iraq -- Operation Iraqi Freedom -- was intended at the outset to be a more focused engagement involving the removal of the Baath Party and its leader, Saddam Hussein, from power and its replacement with a democratic form of government. As the war turned into a protracted counterinsurgency campaign, the notion of "hearts and minds" was resurrected, albeit to extremely limited success. In this regard, the war in Iraq was more similar to the war in Vietnam than to the earlier Operation Desert Storm. In Vietnam, the notion of winning over the hearts and minds of the Vietnamese people, mainly the peasantry who were most vulnerable to intimidation and indoctrination from the Viet Cong, was part of the effort at combating the insurgency -- not the main thrust of the war: the conventional conflict against the North Vietnamese Army.
One can conclude, on the basis of U.S. experience, that winning over the population has far less to do with limited war than with counterinsurgency -- two very different things.