Americans are desperate to avoid another Afghanistan or Iraq in Libya and strongly oppose the use of ground troops. But Americans also seek total victory and the overthrow of Qaddafi. We want to fight a dove's war and win a hawk's peace--and this complicates the endgame for Obama.

Overall, Americans are cautious about the Libyan War. One poll found high levels of support, with 68 percent backing "cruise missile and air strikes in Libya in order to protect civilians from attacks by Qaddafi's forces." But this result may be an artifact of the question's wording, which presses all the right buttons. The evil tyrant Qaddafi is highlighted, but the civil war raging in Libya is not.

Strip these cues away and support for the mission falls significantly. Gallup found that only 47 percent of Americans approved "of the current U.S. military actions against Libya"--while 37 percent disapproved.

There is more evidence of public wariness. When American presidents use force they usually receive a short-term boost in their overall approval ratings, as the public rallies around the leader. But Obama's scores have hardly budged.

And Americans are particularly dovish about the tactics we should employ in Libya, with only 7 percent favoring the use of ground troops.

This wariness is not hard to explain. First of all, we're intervening in the middle of a civil war. For Americans, this is the least attractive kind of military operation. Think Vietnam. Think quagmire.

Second, Libya is not a major threat to the United States. Instead, this is primarily a humanitarian operation. For sure, missions to protect civilians may generate high levels of public approval at first. But this support can quickly evaporate in the face of U.S. casualties or other costs. Think Somalia. Think Black Hawk Down.

Third, in the wake of Afghanistan and Iraq, Americans are sick of war, and eager to focus on domestic issues. In 2009, 49 percent of Americans agreed that the United States should "mind its own business internationally and let other countries get along the best they can." This was the highest figure in over 40 years of asking that question.

But Americans are very hawkish about the objectives of the Libyan War. We're not fighting for a draw. We want to get Qaddafi out.

In a CNN poll on the Libyan War, 77 percent of Americans said it was "very important" or "somewhat important" that Qaddafi be overthrown. In another poll, the figure was 79 percent.

Tapping into this sentiment, Republican Senator Mark Kirk said, "I think that we are now in a shooting war with [Qaddafi]." Therefore, the objective "should be to end his regime and bring about a new government."

This is par for the course in wartime. When we battle an enemy dictator, Americans often favor a crusade for total victory and the tyrant's overthrow. World War II is our model for how war ought to end--with a surrender ceremony in Tokyo Bay.

For example, back in the 1991 Gulf War, President George H. W. Bush fought for restricted goals, to liberate Kuwait. But most Americans wanted to escalate the objectives and remove Saddam Hussein from power.

The dilemma in Libya is clear. The American public won't be impressed by any outcome that falls short of regime change. But they also oppose the use of ground troops that would speed this outcome.

Moving forward, there are many dangerous paths for Obama. But there may be only one narrow course where he gains credit at home. U.S. casualties must be low or zero. The rebels must overthrow Qaddafi. And Libya must avoid collapsing into a failed state or further civil war.

Obama has gambled a great deal of political capital on a rag-tag band of disorganized Libyan rebels that are largely unknown in the United States. The rebels seem to be making military headway, but it's a long way back to Tripoli.

In Operation Odyssey Dawn, Obama will hope to replicate the Iliad, where Odysseus helps win the Trojan War with oratory, diplomatic skill, and guile, rather than the Odyssey, where the Greek hero's exit strategy from conflict in the Mediterranean turns into a long and arduous journey.

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