YAVORIV, Ukraine — The exercise, one of the most fundamental in the military handbook, came off without a hitch. A soldier carrying a length of rope and a grappling hook ran to within 20 feet or so of a coil of concertina wire and stopped.

For a moment, he twirled the rope in his hands like a lasso, then threw the hook over the wire, and tugged hard, testing for explosives.

When nothing happened he signaled two comrades, who ran up and started snipping the wire with cutters.

Although this was a typical training exercise for raw recruits in an elemental soldierly skill, there was nothing typical about the scene. Far from enlistees, these soldiers were regulars in the Ukrainian National Guard, presumably battle-hardened after months on the front lines in eastern Ukraine. And the trainer was an American military instructor, drilling troops for battle with the United States’ former Cold War foe, Russia, and Russian-backed separatists.