On a late-spring day in 2018, when the New York Philharmonic was deep in rehearsals of a Strauss symphony, an unexpected visitor showed up at the stage door of David Geffen Hall, the Philharmonic’s home.

The visitor, Bradley Cooper, the actor and director, had come on a mission. He was preparing to direct and star in a film about Leonard Bernstein, the eminent conductor and composer who led the Philharmonic from 1958 to 1969. He was asking the orchestra’s leaders for help with the movie, “Maestro,” which has its North American premiere on Monday at the New York Film Festival.

The Philharmonic is accustomed to having luminaries at its concerts. But it was unusual for someone like Cooper to express such deep interest in classical music, a field often neglected in popular culture.

“How many top Hollywood stars can be genuine or interested in that way?” said Deborah Borda, the Philharmonic’s then-president and chief executive. “We were really impressed.”

Soon, Cooper was…