PARIS, France — Whoever ends up presiding over Afghan chaos and corruption, Europe is losing patience fast with U.S. mission muddle and a president who won’t make up his mind.

NATO defense ministers meeting last week in Slovakia backed Gen. Stanley McChrystal’s strategy to strike harder, and fast, against a growing insurgency.

But opinion polls and remarks in official circles from Britain to Poland (and elsewhere, like, Canada) make clear that Europeans are fed up with Afghanistan if not the whole military business altogether.

A NATO graph that tracks European defense spending looks like an intermediate ski slope. In every country but Greece, 2010 budget projections suggest a sharper drop.

Jean-Pierre Maulny of France’s Institute for International and Strategic Studies (ISIS) was blunt in the daily Le Monde, which last week examined the trend.

“Budgets are not up to what is at stake, and Europe is sinking into a formless neutrality,” he said. “We are seeing — this is new — a real trans-Atlantic decoupling.”

Europeans spend $520 a year per capita on defense, a third less than Americans. Britain’s budget, the biggest along with France’s, could drop 10 percent in five years.

Where Afghanistan is concerned, these cutbacks coincide with a quickly dissipating political will.

Britain, with 9,000 troops and the largest EU contingent, pledged 500 more but only if allies and Afghans fight the Taliban harder.

Prime Minister Gordon Brown of Britain speaks of Afghanization the way Americans once talked of Vietnamization when looking desperately for lights at the end of tunnels.

Brown told Parliament he was looking for “benchmarks and timelines” to hand over security operations to Afghans. Since 2001, British deaths total 221 — 37 since July.

A Daily Telegraph/YouGov poll in August found 62 percent of respondents wanted troops to come home. In a separate poll, only 27 percent favored a long deployment.

In recent German elections, Chancellor Angela Merkel defeated her Social Democrat opponent, who wants out of the war. She is committed to training Afghan police.

But the influential weekly, Der Spiegel, estimated that 70 percent of Germans, like the defeated candidate, want out of a costly, painful and embarrassing quagmire.

After unleashing world-class blitzkrieg twice in living memory, Germans have mixed feelings about keeping 4,500 soldiers at war. Since 2002, 35 have been killed.

Allies who want more combat support are critical of Germany’s preference for rear-echelon missions. So long after World War II, some say, this is a copout.

In September, a German commander called in an air strike near Kunduz based on sketchy single-source information, which killed innocent Afghans.

Last year, a senior German officer called the program to train Afghan police “a miserable failure,” Tom Coghlan reported in the Times of London.

Later, an official report in Berlin said German troops sit around too much drinking beer and eating sausages.

“Plainly put, the soldiers are too fat, exercise too little and take little care of their diet,” Reinhold Robbe, parliamentary commissioner for armed forces, concluded.

In Italy, feelings flared in September when a Kabul bombing killed six Italians. Crowds demanded that Silvio Berlusconi bring home their 2,800 troops.

Then in October, Coghlan dropped a bombshell in the Times.

In 2008, he wrote, Italian intelligence officers secretly paid the Taliban and local warlords to keep peace in Italy’s sector east of Kabul. No one told French forces who took over and underestimated the threat.

Seeing protection money abruptly cut off, insurgents ambushed a column. They killed 10 Frenchmen, tough Foreign Legion paratroopers, mutilating bodies and seizing weapons.

Though denied in Rome, it was based on solid, if necessarily unnamed, sources.

A senior NATO commander said that while there might have been some reason to pay off insurgents, it was “madness” not to tell a coalition partner.

A “high-ranking Western intelligence source” made the larger point, underscoring the problems of lining up reluctant allies against an ill-defined enemy.

“NATO in Afghanistan is a fragile enough construct without this lot working behind our backs. The Italians have a hell of a lot to answer for.”

France took the news hard. By then, two-thirds of the country already wanted out. But President Nicolas Sarkozy of France, hurrying to Kabul, declared: “My determination is intact.”

That remains the official line, and little is likely to change over the hunkered down Afghan winter. Still, Sarkozy has less reason to stand firm as public will wanes.

In 2007, freshly elected, Sarkozy was eager to show strength to Washington, lining up as a solid ally after Jacques Chirac’s iffy relations with George W. Bush.

With Barack Obama, Sarkozy is shifting back toward France’s usual Gaullist role: an independent-minded, nuclear-tipped force capable of taking its own direction.

In London, Gen. David Richard, chief of the General Staff, summed up the allies’ challenge across the continent in remarks to the Daily Telegraph.

Holding public support is a struggle, he said. “We need to do better.”