President Barack Obama took some Republican senators to dinner recently, and then gave Congress the unusual courtesy of four presidential visits to Capitol Hill.

Can a flurry of presidential schmoozing change Washington from a polarized battlefield into a haven of bipartisan cooperation? Don't hold your breath.

"We're very far apart," an Obama aide told me, stating the obvious in the wake of the budget proposal from fiscal hawk Rep. Paul D. Ryan, R-Wis. The GOP wish list includes deep cuts in health care, education and defense spending, but no tax increases. The Democratic budget, released Wednesday, includes more modest spending cuts and almost $1 trillion in tax increases.

So why the charm offensive? It is the byproduct of an important change: Both sides are older, sadder and wiser.

Republicans have finally noticed that they lost the presidential election and that most voters didn't choose their severely conservative policies. At best, Ryan noted in his budget announcement, the voters opted for "divided government. So this year, we have to make it work."

Democrats have suffered a comeuppance too: Voters didn't rise up in anger against the automatic "sequester" of government spending that kicked in March 1, as liberals hoped they would. That may be because most of the sequester's effects haven't been felt, but the short-term result was to make Obama and his party look silly - and, more important, to knock some points off the president's popularity.

There's been gradual movement toward the center on the substance of the budget. Republicans still oppose tax increases, but some senators have acknowledged they'll have to accept revenue increases to make a deal. And Obama has been more explicit than before about his willingness to cut future spending on Medicare and Social Security. The more the president repeats that message to Democrats, the more Republicans believe he might mean it.

And leaders on both sides know how to get to an agreement - or how not to get there - because they've tried so many routes before.

Negotiations between Obama and House Speaker John A. Boehner, R-Ohio, won't work; that failed miserably in 2011. Negotiations with Senate Republican leader Mitch McConnell, R-Ky., produced a short-term deal in 2012 but probably won't work now because McConnell, who faces a primary challenge next year, has been adamant in his opposition to new taxes.

That leaves only a bipartisan negotiation among senators who want to negotiate, like the "Gang of Six" who produced a budget compromise in 2011 (a deal that led nowhere at the time).

"The only way we've been able to get anything done over the last two years is when the Senate passes something on a bipartisan basis and sends it to the House," the Obama aide noted. Only then has Boehner allowed his House majority to divide and contribute to passing legislation.

That's why Obama kicked off his charm offensive with senators - and not just any senators but Republicans who were already interested in seeing whether a deal is possible, including some fiscal moderates (Lindsey Graham of South Carolina, Kelly Ayotte of New Hampshire) and influential fiscal hawks (Pat Toomey of Pennsylvania, Tom Coburn of Oklahoma). If it didn't change the landscape, it may have detoxified the air.

"I think the president's tremendously sincere. I don't think this is just a political change in tactic," Coburn said on NBC's "Meet the Press" last week. "I think he actually would like to solve the problems of the country. … He's moving in the right direction. I'm proud of him for doing it." When was the last time you heard a Republican say something like that about Obama?

Obama needs the vote of confidence.

Beyond the budget, he is hoping to pass new legislation on immigration and gun control. "(If) we can build these bonds of trust, that will be useful on these other issues," the Obama aide told me.

And despite Coburn's disclaimer, there's the ultimate political calculation: Who will voters blame?

"There's a tendency to say, whenever agreement isn't reached, that it's both sides' fault," the aide said. "If we reach out and we put entitlements on the table and are serious about it, and the Republicans still can't get to 'yes,' there is no other answer left other than Republicans just can't get to 'yes.'"

So don't get too romantic about all the wining and dining in Washington. It won't mean an end to the battles; it's more a cease-fire while both sides warily regroup. But it's better than nothing, and a lot better than everyone going home hungry.

Readers can email Doyle McManus at doyle.mcmanus@latimes.com.