Dan O’Brien’s ‘‘The Body of an American’’ and Robert Schenkkan’s ‘‘All the Way’’ are the inaugural winners of a new theater award honoring the late Senator Ted Kennedy. Columbia University announced Friday the two playwrights will share the award — and the $100,000 prize. Officially called The Edward M. Kennedy Prize for Drama Inspired by American History, the award honors a new play or musical exploring US history and issues of the day. Schenkkan’s play begins in November 1963 with Lyndon B. Johnson’s rise to the presidency after the assassination of Kennedy’s brother John F. Kennedy, and ends a year later with Johnson’s win over Barry Goldwater. O’Brien’s play was inspired by the experiences of Paul Watson, who won a Pulitzer for a 1993 photo of a dead US Army Ranger dragged through the streets of Somalia’s capital.