Two presidential campaigns dealing with sudden reversals of fortune descended on this must-have state Tuesday, one hoping to sustain a new momentum, the other hoping to regain its footing.

In both rhetoric and demeanor, President Obama and Republican Mitt Romney came to Ohio acutely aware of the altered terrain.

Romney, buoyed by new polls that show him pulling ahead of the president, has shed the languid pace that characterized his travels as recently as last weekend. He appears renewed, even ebullient, and so do his crowds. About 500 people endured a driving rainstorm and a muddy field to see him Monday evening in Newport News, Va., and an estimated 12,000 packed a parking lot here for a Romney rally Tuesday night.

Obama, speaking to a sprawling rally at sunset on the Ohio State campus Tuesday, made it clear that he understands the stakes. He urged everyone who had not registered to vote to do so on his campaign Web site by Tuesday’s 9 p.m. deadline.

“I know it’s easy to procrastinate in college,” he said. But now, he added, “no explanations, no excuses. . . . Everything we fought for in 2008 is on the line in 2012.’’

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The battle has taken on new urgency after Romney’s widely praised performance in the first presidential debate last week — combined with Obama’s lackluster showing — eroded Obama’s lead here and in some national polls.

One survey released Monday by the Pew Research Center showed Romney ahead by four percentage points among likely voters nationwide, a 12-point swing from Pew’s mid-September survey. That was followed Tuesday by Gallup’s first survey of likely voters, which showed Romney leading Obama by 49 to 47 percent, although the president retained an edge among registered voters.

But the race comes down to a few crucial battleground states, none more important than Ohio. And Romney has struggled here. One measure taken after the debate, a CNN/ORC International Poll released Tuesday, showed Romney down by four points in Ohio, though the race has tightened here, as well.

No Republican in the modern era has been elected president without winning the state, and for the moment, the former Massachusetts governor faces significant challenges in trying to win Ohio’s 18 electoral votes. Unemployment stands at 7.2 percent, lower than the national average of 7.8 percent.

Republicans argue that Gov. John Kasich (R) is more responsible than Obama for the rosy picture, and the debate over who should take credit for better times has muddied Romney’s message that new leadership is needed to fix a bad economy.

“There’s a battle over the narrative on the economy here,” said Greg R. Lawson, a policy analyst at the Buckeye Institute, a conservative think tank based in Columbus. “If you think it’s the president or if you think it has much more to do with the Kasich administration. . . . The economy and that narrative will be decisive.”

The Obama administration, taking nothing for granted in a state it desperately wants to win, has been showering Ohio with attention and money throughout the president’s term. Excluding the District’s neighboring states, Virginia and Maryland, Obama has visited Ohio more than any state except New York, where he often travels to raise money.

The auto bailout that the administration oversaw also plays well for Obama in Ohio, home to a number of important GM and Chrysler factories and ancillary auto parts plants.

The Obama camp said it is prepared to defend its advantage in the state. “Ohio is Obama country,” campaign spokeswoman Jennifer Psaki said Tuesday, adding that the contrast between the president and Romney on the economy is in boldest relief here. That message, she said, is backed by a campaign organization that never really left the state after 2008 and just opened its 120th field office.

Whatever the obstacles in Ohio, Romney — known for his occasionally wooden demeanor on the campaign trail — has looked in recent days as if he is enjoying himself. On Monday, after finishing a foreign policy speech at the Virginia Military Institute in Lexington, he broke with the campaign’s lockstep schedule and turned his motorcade around for an unplanned stop to greet hundreds of children lined up and waving outside Fairfield Elementary School. Joined by Virginia Gov. Robert F. McDonnell (R), he grinned and laughed as the children began running toward them.

Romney was still smiling minutes later as he brought two oversize McDonald’s bags with him on a foray to the back of his campaign plane, handing out food to reporters. “Quarter-pounder? Fries? Chicken sandwich?” he called out.

Romney aides made clear Tuesday that they will not rest in a race that has swung back and forth.

“This is a campaign that has never gotten too high when things are good and too low when things are bad,’’ said Kevin Madden, a senior Romney adviser. “The governor, in particular, remains very focused on the task at hand, which is making sure he talks directly to those voters who haven’t made up their mind yet. . . . I think if he continues to do that, we’re going to be well positioned in this race.’’

Romney’s bounce in the polls confirms some things that the campaign’s research was showing even before the debate — chiefly, that voters want to hear more directly from the candidate himself. That is why, for instance, the campaign has released its first television ads in which Romney speaks directly to the camera.

Campaign strategists are taking other steps, as well, to build on their momentum. They are retooling Romney’s stump speech so it has fewer red-meat lines that aim to stir up conservative partisans who attend rallies and more points that are intended to appeal to persuadable voters who will be seeing and reading snippets from the speech in news coverage.

“We want to capitalize on what we saw in the debate, what worked in the debate — and it has the added benefit of being who he is,” said Ed Gillespie, a former Republican National Committee chairman who has joined the campaign to bring more clarity to its message.

Other new measures include a revival of the tele-town hall meetings in which the campaign robo-dials thousands of supporters in a swing state and Romney holds a town hall meeting via conference call. Romney did such conference calls often during the Republican primaries and has very recently begun doing them again, including sessions with Virginia and Colorado voters in the past few days while he was in other states.

Romney is also emphasizing a recent campaign strategy to clearly spell out what he would do as president, as he did when he introduced his vision for a “vibrant rural America” at a campaign stop Tuesday outside Des Moines.

At the rally, held in an open soybean field next to a giant grain silo, some of the 1,200 people in attendance praised Romney’s debate performance and said they weren’t surprised by his recent momentum.

“There was no comparison in that debate, it was like night and day,’’ said Sally Cole, 55, who drove 3½ hours from her home in New Hampton, Iowa, to see the Republican nominee. “Obama seemed annoyed, disgusted, preoccupied and befuddled. Romney struck me as respectful, assertive and informed.’’

Obama aides continue to say that they have always expected a tight race and have tried to run every day as if they are five points down. In that sense, the campaign has remained pretty much like its candidate: outwardly placid and businesslike, no matter the turbulence underneath.

At the Ohio State rally Tuesday, the president urged Ohioans to reject a return to the Republican economic policies that he said caused the financial crisis.

“We love you,” a voice rang out.

“I love you back, but I need you to vote,” Obama said.

Philip Rucker in Boston, Karen Tumulty in Washington and Rosalind S. Helderman in Ohio contributed to this report.