It seems ludicrous on some levels to compare the Austrian-born, former bodybuilding, movie-star governor to the new president, whom Schwarzenegger once mocked for his “skinny legs” and “scrawny little arms.”

The governor’s Republican Party hasn’t controlled either chamber of the state legislature during his tenure, and the party’s conservative base has long disdained his pro-choice, pro-environmental regulation and pro-social spending stances.

Yet the fundamental lesson of Schwarzenegger’s California experience offers a cautionary tale for Obama: A swooning economy drags down even the most popular leaders.

Republicans can talk all they want about tax-and-spend Democrats, about Cabinet secretary nominees not paying taxes and about not wearing jackets in the Oval Office. If the economy rebounds in a meaningful way over the next few years, Obama’s talk of change will resonate, and his star will shine ever brighter.

On the other hand, no amount of mellifluous prose, Internet savvy or bipartisan appeal will overcome an economy in which plunging stock prices ruin retirements, bank failures dry up credit and one in 10 Americans is out of work.

“The forces of gravity in politics are strong,” says Rob Stutzman, who served as Schwarzenegger’s chief communication strategist during his first several years in office. “Someone comes in on the promise of change, and people expect things will change.”

Americans tell pollsters they understand the dire state of the economy and will give Obama time to turn things around. Schwarzenegger’s fall from grace shows the limits of the public’s patience.

When the former Mr. Universe took office, he elicited eye-popping stares in public, even from journalists and seasoned politicians. Schwarzenegger was so popular in his first year as governor that a significant plurality of San Francisco-area voters — a region that has not elected a Republican to Congress in more than a decade — gave him a thumbs-up.

His bipartisan appeal was so strong that he easily won re-election in one of the nation’s bluest states, even after trying to make an end run around Sacramento Democrats the year before with a series of ballot initiatives that met with disastrous results.

But his popularity is heading in the opposite direction of California’s unemployment rate, which stood at 9.3 percent in December, the fourth-worst in the nation, and is expected to exceed 10 percent when new figures are released Wednesday, the highest rate since President Reagan’s first term.

Of the nation’s 10 worst metropolitan areas for home mortgage defaults, eight are in California. The state’s bond rating is now the second-worst in the nation. And 200,000 state workers took unpaid leave Friday, a forced twice-a-month furlough aimed at easing a $42 billion deficit.

Schwarzenegger’s approval ratings, which once hovered at Obama-like levels, have fallen to record lows. The latest survey from the Public Policy Institute of California showed that fewer than one in three approve of Schwarzenegger’s handling of the state’s economic troubles. More than half disapprove of the job he is doing as governor.

Ironically, Schwarzenegger is more ideologically in tune with Obama than many Democrats.

He praised the president when he allowed California to tighten automobile emissions, cheered the president when he expanded health care for children and joined Obama’s call on Congress to quickly pass the stimulus bill. He even had exclusive seats at the inauguration.

Such “post-partisanship” is what is needed for a Republican to win in a state like California. But it has hardly helped his popularity. Schwarzenegger will leave office after 2010 because of term limits, but voters would have given him the heave-ho regardless.

The same PPIC survey showed that residents of the nation’s most populous state have high hopes for the new president. More than eight in 10 said they were optimistic about Obama’s chances to turn the economy around.

As Schwarzenegger knows, Obama will pay a price if he doesn’t.