It's getting tougher for the U.S. to impose its will, but we can still lead the world -- the trick is convincing the world to follow.

Here are a few of the big, global problems that the U.S. has recently tried and failed to resolve:

• North Korea's recent test-launch of a long-range missile, which U.S. diplomacy and threats couldn't deter.

• A new war between the Sudans, breaking a short-lived peace that the U.S. spent years building.

• Syria's continuing massacre of civilians, for which neither American diplomats nor American generals can find an acceptable solution.

• Egypt's tightening military rule, which has gotten so bad that the U.S. spent weeks just to extricate some detained American NGO workers.

• Israel's settlement growth in Palestinian territory, which the U.S. opposes as a barrier to Middle East peace.

• The Yemeni president's refusal to abdicate power, despite a U.S.-brokered pledge that he would step down.

• Afghanistan's unceasing war with itself, to which ten years of American-led war have not brought peace.

• Iran's nuclear development, which looks to be continuing despite U.S. sanctions and recent U.S.-led disarmament talks.

The U.S. isn't powerless. It's significantly alleviated most of these conflicts, and it's taken the international lead on all of them. But the pattern is unmissable. It is a big, complicated world in which the U.S. can only do so much. We're the most powerful country in the world by far, but that doesn't always make us the bosses. This might seem obvious, but American domestic discourse -- not to mention foreign discourses -- often seem to assume a strength of American hegemony that just doesn't exist.

President Obama's major foreign policy addresses, like those of the presidents before him, take American dominance in world affairs as both necessary and absolute. There's nothing wrong with declaring that Iran will not be allowed to build a nuclear weapon or that democracy will come to the Middle East. And there's nothing wrong with the American leader discussing those issues from an American perspective. After all, the U.S. is the strongest and richest country in the world, which also makes it the best positioned to help. But there's a difference between helping and solving, just as there's a difference between offering leadership and having others follow. We seem to assume the latter (as do many non-Americans, for example in Egypt, where it's common to assume "foreign hands" guide Egyptian politics when in fact the U.S. seems to have less influence there every day), imagining American power extends far beyond its actual limits.

Part of this is domestic politics. Mitt Romney was probably making a smart political move to jump on Obama's hot mic comments to Russian President Dmitri Medvedev about how the U.S. couldn't make another nuclear arms reduction deal until after the election. Reducing American might is politically unpopular (even though we don't actually need those thousands of nuclear warheads) as is the idea of offering concessions to another, not-so-friendly country. It would be bad politics for Obama to enter tough and maybe even painful negotiations with a competing nation, probably because this conflicts with the Reagan-era idea that America's inherent strength and goodness means that we dictate terms to the world. But even Reagan compromised and horse-traded with Moscow, though he also had the good sense not to do it during an election.

This is the big conflict between how U.S. leaders negotiate American politics and American foreign policy: the former requires confidence, the latter humility. But the two are not inseparable. Maybe because our political system promotes leaders who believe most strongly in American power, or maybe because it pressures those leaders to exercise more power than they might actually have, it can often seem that the U.S. is constantly falling short of our ambitions. We can't stop Israeli settlement growth, Iranian nuclear development, Sudanese civil war, AIDS in Africa, or terrorism in Pakistan, even though Americans presidents keep insisting that we will.

There was a time when we seemed to have more influence on how other countries behaved. In this 1980 map of Cold War alliances, the "blue" countries would reliably, if not always, follow U.S. leadership. Part of that was because we had easier requests then; it's one thing to tell Pakistani generals to train anti-Soviet fighters, quite another to ask them to give up power to democratic institutions. But the threat of Soviet domination gave us a common mission that made cooperation more attractive and American leadership more desirable. There's no more great red menace to unify the majority of the world under American leadership. Other countries don't need us in the way that they used to.

The good news is that American and global interests still tend to line up pretty frequently. That's not a coincidence. The U.S. does more than any other country at maintaining global peace, cooperation, and free trade. The rest of the world might not depend on American protection from the Soviet Union, but it depends on the U.S.-enforced political and economic order. That's the new American leadership. When China slashed its Iranian oil imports by half -- a big blow to Tehran and a boost to the U.S.-led effort to isolate Iran -- it wasn't because Obama called up Chinese President Hu Jintao and told him to do it. The U.S., through a lot of difficult and sometimes painful diplomatic and economic maneuvering, found a way to line up American and Chinese interests.

This sort of power makes the U.S. good at promoting democracy, cooperation, and free trade -- Burma's opening, for example, or China's remarkably peaceful rise -- but less effective at stopping civil wars or convincing dictators to do things that might threaten their own rules (or lives). If Iranian leaders believe they need a nuclear program to save themselves from a U.S. invasion, they're going to keep it. And the logic of ethnic conflict or religious terrorism can't really be refuted by, say, American trade incentives.

When U.S. interests line up with global interests, we suddenly become very effective at leading the world: isolating Iran, convincing Sudan to allow its southern third to secede, or curbing Chinese trade abuses, for example, would probably all have been impossible on our own. But they also wouldn't have happened without the U.S. taking the lead. That means that U.S. leadership is becoming more about finding opportunities for cooperation and compromise than it is about, say, the strength of our military or force of our ideas, although those help too. Sometimes the U.S. president has to tell his Russian counterpart that he'll offer some concessions in exchange for, say, dismantling Soviet-era nuclear weapons or reducing arms sales to Syria. That's not a particularly jingoistic vision of American leadership, and it's not likely to play well in a political campaign. But that's the world we live in.