On both sides of the Atlantic, the ties that held the alliance together are weakening. On the American side, Obama’s biography links him to the Pacific and Africa but not to the old continent. His personal story echoes the demographic changes in the United States that have reduced the influence of Americans of European origin.

Meanwhile, on the European side, the depth of the euro crisis has crowded out almost all foreign policy from the agenda of Europe’s top decision-makers. The end of the Cold War means that Europeans no longer need American protection, and the U.S. financial crisis has led to a fall in American demand for European products (although U.S. exports to Europe are at an all-time high).

What’s more, Obama’s lack of warmth has precluded him from establishing the sorts of human relationships with European leaders that animate alliances. When asked to name his closest allies, Obama mentions non-European leaders such as Recep Tayyip Erdogan of Turkey and Lee Myung-bak of South Korea. And his transactional nature has led to a neglect of countries that he feels will not contribute more to the relationship — within a year of being elected, Obama had managed to alienate the leaders of most of Europe’s big states, from Gordon Brown to Nicolas Sarkozy to Jose Luis Rodriguez Zapatero.

Americans hardly remember, but Europe’s collective nose was put out of joint by Obama’s refusal to make the trip to Europe for the 2010 EU-U.S. summit. More recently, Obama has reached out to allies to counteract the impression that the only way to get a friendly reception in Washington is to be a problem nation — but far too late to erase the sense that Europe matters little to this American president.

Underlying these superficial issues is a more fundamental divergence in the way Europe and the United States are coping with their separate declines. As the EU’s role shrinks in the world, Europeans have sought to help build a multilateral, rule-based world. That is why they, rather than the Chinese or the Americans, have pushed for the creation of institutionalized global responses to climate change, genocide and various trade disputes. To the extent that today’s world has not collapsed into the deadlocked chaos of a “G-zero,” it is often due to European efforts to create a functioning institutional order.

To Washington’s eternal frustration, however, Europeans have not put their energies into becoming a full partner on global issues. For all the existential angst of the euro crisis, Europe is not as weak as people think it is. It still has the world’s largest market and represents 17 percent of world trade, compared with 12 percent for the United States.

Even in military terms, the EU is the world’s No. 2 military power, with 21 percent of the world’s military spending, versus 5 percent for China, 3 percent for Russia, 2 percent for India, and 1.5 percent for Brazil, according to Harvard scholar Joseph Nye. But, ironically for a people who have embraced multilateralism more than any other on Earth, Europeans have not pooled their impressive economic, political and military resources. And with the eurozone’s need to resolve the euro crisis, the EU may split into two or more tiers — making concerted action even more difficult. As a result, European power is too diffuse to be much of a help or a hindrance on many issues.