When Arnold Schwarzenegger takes the stage tonight at the Republican National Convention, he will undoubtedly dazzle the delegates, but it may be best for him if his performance is something short of a showstopper.

While the movie star turned California governor is riding high in polls taken in his home state, President Bush’s popularity among voters there is at a low ebb. Tonight, the 57-year-old former actor faces the daunting challenge of trying to transfer some of his mojo to the president without doing any damage at home.

“I think Arnold is so shrewd politically that he’ll calibrate his approach very, very carefully,” said a California-based Republican political strategist, Arnold Steinberg.

Mr. Schwarzenegger’s aides have given few hints about what he will say tonight. They have said he plans to underscore the Republican Party’s commitment to economic opportunity by pointing to his history as an immigrant from Austria who found great success in America.

“There is an ego involved here and it is important for him to legitimize what he has done as governor,” said a political analyst at the University of Southern California, Sherry Bebitch Jeffe.

The governor is expected to steer clear of a variety of social issues – such as abortion and the proposed constitutional amendment to ban gay marriage – on which he disagrees with both Mr. Bush and the party platform.

It may surprise most Americans to hear Mr. Schwarzenegger, known for his muscular presence in high-octane action films, described as a deft and cautious politician. After all, the last time he found his way into the national media spotlight he was engaged in a knockdown, drag-out battle to win the California governor’s job through an unprecedented recall election.

Mr. Schwarzenegger prevailed in that fight last fall, but only after a white-hot press frenzy over sordid allegations of repeated sexual misconduct toward women during his two and a half decades in the movie business.

In his first nine months in office, the former bodybuilding champion has won the respect of lawmakers, political analysts, and the press for rolling up his sleeves and mastering the details of the state’s budget woes and other problems.

“I think everyone was surprised that he was that quick of a political study,” said a communications professor at California State University, Barbara O’Connor. “He’s very focused and this far he has not deviated from his agenda. He had a list when he came in. He went right down that list, some of which he had success with, some he didn’t,” she said.

After denouncing the Democrats who control the state legislature as “girlie men,” Mr. Schwarzenegger managed to reach a budget deal. Critics have accused him of papering over the budget gaps with fiscal gimmicks remarkably similar to those used by the unpopular Democratic governor he unseated, Gray Davis.

“He’s had some important political triumphs in California, but there are also some significant political concessions that perhaps shouldn’t have been made,” Mr. Steinberg said.

In 2006, Mr. Schwarzenegger will be up for re-election. It seems a fair bet that he will run, though some note that the state’s fiscal problems are likely to re-emerge at about that time. There is even buzz about a possible bid for the presidency in 2008.

At the moment, the Constitution bars Mr. Schwarzenegger from holding the top office because he was not born in America. But for a man who tossed out the sitting governor of America’s most populous state, no obstacle is insurmountable.

While coyly insisting that he has given no thought to seeking the presidency, Mr. Schwarzenegger has said that he favors changing the Constitution to allow immigrants to run for president. A proposed constitutional amendment on the subject has never made it out of Congress.

“I’m not at all sure it will be in time for him to take advantage of it,” said Ms. Jeffe. She said a Senate bid by Mr. Schwarzenegger is a more likely prospect.