As Gustavo Dudamel looks out over the Los Angeles Philharmonic, only his eyes are in motion. They settle briefly on the faces of each of his 100 musicians. There is a moment of charged silence, a collective inhalation. Then he raises his baton with the life-or-death urgency of a rocket pilot just before liftoff. His face, his mien, his whole body seem to be saying, “I am going to take you on an adventure.” No matter how often the orchestra members may have played this piece before, Dudamel’s bearing indicates that this performance really matters, that there are truths yet to be disclosed.

Suddenly his hands come to life. The baton in his right slices through the air, dividing the score into beats and patterns. His left—fluid, graceful—shapes the tone and soul of the playing. In softer passages he holds the ensemble back with carefully restrained, exacting gestures that permit only small fractions of sound to escape into the recesses of Disney Hall. In more lyrical and expansive sections he relaxes visibly, leans back, swings his arms as though to embrace a cosmos. After a particularly beautiful solo, Dudamel beams at the performer with delight. Watching him is like reading a map of the score.

Yes, his long, curly hair can sometimes bounce around like a sideshow. But for all his youthful energy and enthusiasm, the 30-year-old music director of the L.A. Phil is not playing to the audience. This is not the classical music equivalent of air guitar: Dudamel knows that most of the work behind any successful concert has already been accomplished in rehearsal, as passages are played again and again, started, stopped, and polished until they shine. Now his duty is sending “real time” signals to his musicians, not dancing for the crowd.

Dudamel never wants for authority. He has been leading orchestras since he was a teenager and has been world famous for more than a quarter of his life. Still, his abiding emotion seems to be one of gratitude—to the pl...