“We have to figure out how to adapt to this new environment,” said Col. Clair A. Gill, a 1994 West Point graduate who commands the 10th Combat Aviation Brigade from Fort Drum, N.Y., which flies Apache, Black Hawk and Chinook helicopters. “I don’t think we’re there yet.”

To be sure, commanders are expected to argue that the wars in Iraq and Afghanistan will still require thousands of American troops for the foreseeable future. But the United States and its NATO allies recently completed positioning about 4,500 soldiers in the three Baltic States and Poland, and are preparing to keep several thousand armored troops on the Continent as a deterrent to Russian aggression.

These tensions are part of an expanding rivalry and military buildup, with echoes of the Cold War, between Washington and Moscow.

Moscow is flowing forces for its own exercises along its western border with Europe and also deploying in Syria and eastern Ukraine, and is building up its nuclear arsenal and cyberwarfare prowess in what American military officials say is an attempt to prove its relevance after years of economic decline and retrenchment. Russia has scheduled a major exercise in September that could involve as many as 100,000 troops and other security personnel.