The cast of “Hamilton.” (Joan Marcus)

Thomas Jefferson was honored with a majestic memorial, George Washington by the name of our capital city. But lovers of show tunes know that among the Founding Fathers, Alexander Hamilton has received the highest accolade of all: his own musical.

And not just any musical. A great musical. “Hamilton,” which had its official opening Thursday night at Broadway’s Richard Rodgers Theatre, is a captivating mirror of the man whose life it surveys: blazingly original, restlessly innovative, magnetic from start to finish.

The trip from off-Broadway’s Public Theater, where it opened in February to rapturous notices, has been good for “Hamilton,” written entirely by the young maestro in its title role, Lin-Manuel Miranda. Under Thomas Kail’s superb direction, the 21 cast members have deepened their performances, and several — including Leslie Odom Jr., Renée Elise Goldsberry, Daveed Diggs and Phillipa Soo — now offer up even sharper powerhouse turns than they did downtown.

The Rodgers stage seems to provide a fresh boost, too, to the dozen or so dancers who perform choreographer Andy Blankenbuehler’s muscular, staccato steps with even more sensuous relish than before. Executing everything from hip-hop moves to military drills to martial-arts kicks, they’re the sultry cylinders propelling the show’s creative engine inexorably forward — a physical expression of the breakneck speed at which Hamilton vaulted through life. Not for nothing does the first act close with a number titled “Non-stop.”

“I’m not throwing away my shot!” Miranda-as-Hamilton sings again and again, delivering in syncopated rap the lyrics that become a sort of motto for an immigrant rebel ever mindful of his humble origins, as well as the idea of America at its own birth as a land of exhilarating new beginnings. The declaration is a foreshadowing, too, of Hamilton’s ending, coming for him on a bluff overlooking the Hudson River, out of the barrel of Aaron Burr’s dueling pistol.

Daveed Diggs (center), Anthony Ramos, Carleigh Bettiol, and Thayne Jasperson in “Hamilton.” (Joan Marcus/Joan Marcus)

That Miranda, composer of the Tony-winning hit “In the Heights,” writes his new revolutionary pageant in an urbane musical vernacular for a cast of actors of every color gives thrilling universal textures to a story that could have become message-laden or starchy. That never comes close to happening. “Hamilton” is American history as the hippest course in the curriculum.

Running through this project there is, in fact, the marvelous affirmational conviction of what the United States was meant to be. As you experience nation-building through the eyes of Miranda and company you can’t help but feel a pang of regret over the toxically divided country that we have become. Then again, “Hamilton” — based on Ron Chernow’s well-received biography “Alexander Hamilton” — does crack open the events of 1776 and beyond to reveal the bitter clashes of personality and ideology that convulsed the founders. The making of America was anything but a sedate party debate.

This makes the ingenious, egocentric, abrasive Hamilton a terrific touchstone for an evening of musical theater. “Hamilton” is indeed a great epic, tracing the successes and failures of a trenchant political thinker and flawed family man. He is not a man of self-control: He cheats scandalously on his angelic wife, Eliza (Soo, in an elegant portrayal), and chafes childishly under the command of Washington (a bracingly effective Christopher Jackson). He flirts brazenly with the affections of Eliza’s sister Angelica (Goldsberry, in an utterly enchanting turn) and, most disastrously, makes a relentless enemy of his political rival, Burr, played here with exceptional finesse by Odom. His Burr is no two-dimensional villain; he’s just too much of a pragmatist for a true believer like Hamilton to stomach. Their mutual contempt metastasizes into mutual tragedy.

Miranda’s own performance feels weighty without being showy; it’s generous to his sparkling castmates, who also include such standouts as Okieriete Onaodowan, Anthony Ramos and Jasmine Cephas Jones. Diggs portrays the Marquis de Lafayette in Act 1 and, in Act 2, a wondrously preening version of Jefferson, returning from Paris after the war to sing “What’d I Miss.” Even more delicious is Jonathan Groff, decked out in costume designer Paul Tazewell’s glittery royal finery as King George III. He saunters onto David Korins’s splendidly rustic set from time to time to sing dismissively about his rebellious subjects’ misguided adventurism, in the delightful “You’ll Be Back.”

With input from Kail, Blankenbuehler and music director-orchestrator Alex Lacamoire, the 30-odd numbers, an exciting range of pop ballads and hip-hop operettas, take us enthrallingly from Hamilton’s public world to his private life and back again. Some of the best: “Helpless,” a divinely refined girl-group treatment of Hamilton’s courtship of Eliza, through her eyes and Angelica’s; “The Room Where it Happens,” a bluesy elucidation of a politician’s urge to be at the center of the action; “Yorktown,” a dramatically potent dance number positing the war’s end as a sinewy ballet.

The intense pride you feel on an evening such as this, when America’s story is retold with such style, verve and imagination, is not of a sort that’s come by easily on Broadway. There’s a magnitude of joy summoned here that suggests that King George is absolutely right. You most certainly will want to be back.

Hamilton Book, music and lyrics by Lin-Manuel Miranda. Directed by Thomas Kail. Choreography, Andy Blankenbuehler; set, Thomas Korins; music direction, Alex Lacamoire ; costumes, Paul Tazewell; lighting, Howell Binkley; sound, Nevin Steinberg. With Okieriete Onaodowan, Anthony Ramos, Christopher Jackson. About 2 hours 45 minutes. At Richard Rodgers Theatre, 226 W. 46th St., New York. For more information, visit www.ticketmaster.com or call 877-250-2929.

Jonathan Groff is priceless as King George III in “Hamilton.” (Joan Marcus)