Inside the Edward M. Kennedy Institute for the United States Senate, there’s an exact replica of the chamber. Outside, beneath a plastic tent shaking in the chilly wind coming off Boston Harbor, the Washington political world descended Monday, laughing and tearing up, telling their tales of the Massachusetts senator they remembered as the embodiment of a Senate now all but gone.

The stories were vivid and, in many instances, sentimental. But the message was anything but: If today’s senators want to be remembered for anything more than their fractiousness and use of procedural tools to block legislation, they have to start developing more of a Kennedy-esque rapport — among themselves and with the institution.

Sen. John McCain (R-Ariz.) remembered a proxy screaming match on behalf of two freshmen that finished with Kennedy teasing McCain about them showing off how real senators brawl.

Former Rep. Patrick Kennedy (D-R.I.) told the audience that he could almost see his father in the crowd, standing in the cold, making sure everyone got a cup of coffee.

Vice President Joe Biden remembered when he first arrived in Washington in 1973, Kennedy introducing him to Republican senators by bringing him to the Senate gym. Biden was clothed, most of them were naked, Biden said, and “Oh God, was I embarrassed.”

And President Barack Obama, whose path to the presidency began in this city with his keynote address to the Democratic National Convention 11 years ago, and whose endorsement by Kennedy in 2008 essentially sealed him the Democratic nomination, recalled a man who worked with Sen. Orrin Hatch to pass the Children’s Health Insurance Program. Kennedy’s technique: He alternated between having his chief of staff serenade the Utah Republican and puffing on a cigar to annoy him.

Since the midterms, especially, Obama has taken a dismissive approach to Congress, feeling no longer constrained by making a show of reaching out to Republicans, who, he’s learned, don’t want to work with him anyway. From his immigration-reform executive actions to his plan for moving on any kind of deal that would come out of the Iran talks, White House aides don’t have much interest in thinking about the Senate or the House anymore in planning Obama’s agenda.

On Monday, though, Obama was back to talking about cooperation and bipartisanship.

“What if we carried ourselves more like Ted Kennedy? What if we worked to follow his example a little bit harder?” Obama said. “People fight to get in the Senate, and then they’re afraid. We fight to get these positions and then don’t want to do anything with them. Ted understood the only reason to get these positions is to get something done.”

En route to Boston, Obama’s spokesman Eric Schultz put it in combative terms more typical for the White House of late.

“We’re troubled that Congress in this day and age, especially as we head to the Edward Kennedy Institute for the United States Senate, both struggles to fund the Department of Homeland Security and now cannot pass a bill discouraging human sex trafficking.”

Kennedy waited a year to deliver his first speech on the Senate floor, Obama recalled at the institute, noting dryly that “that’s no longer the custom.” (Freshman Arkansas Sen. Tom Cotton had barely been sworn in before he riled up the White House over his own maiden speech and his open letter attacking the Iran talks.) The president looked over to former Senate majority leaders Trent Lott and Tom Daschle, joking that they could talk about the time when traditions meant something, though he left out his own impatient ambition that led him to announce a presidential campaign two years into his first term.

Obama spoke instead of his own arrival in the Senate 10 years ago, opening up a desk that had been carved with the names of previous occupants — Robert Taft, Howard Baker, Paul Simon, Paul Wellstone and Robert F. Kennedy. Being in the Senate, he said, made him want to be more serious about policy and bipartisanship, and the memory of Ted Kennedy is “the essence of what it can be.”

“It’s a more diverse, more accurate reflection of America, and that is a grand thing, a great achievement, but Ted grieved the loss of camaraderie and collegiality,” Obama said. “I think he regretted the arguments now made to cameras, instead of to colleagues, directed at a narrow base instead of the body politic as a whole, the outsize influence of money and special influence, and how it all leads more Americans to turn away in disgust and not exercise the right to vote.”

McCain, who was repeating a few stories from his 2009 Kennedy eulogy as Obama arrived, mused about his old fights with Kennedy.

“I knew the Senate wouldn’t be the same without him. And it hasn’t been. That’s mostly for reasons that have nothing to do with Ted,” McCain said, “but I have no doubt the place would be a little more productive and a lot more fun if he were there.”

Nearly 2,000 Kennedy colleagues, family members, friends and former staffers were here for the ceremony. There wasn’t even anyone to greet Obama when Air Force One arrived at Logan Airport because all the local officials were already at the ceremony — which included Biden doing a ceremonial gaveling in of the institute’s replica of the Senate chamber.

“He was the anchor of our personal lives and he was the anchor of the Senate as an institution,” Biden said.

Obama stopped by the re-creation of Kennedy’s Senate office in the institute, and in his speech, recalled Kennedy showing him around. He also made an unannounced visit to the replica of the Senate chamber and made some off-the-cuff remarks about the importance of service.

Obama finished his short day in Boston in another place important to him, at a Democratic National Committee fundraiser in Cambridge not far from where he attended Harvard Law School.

Years later, the connection between the Kennedys and Obama still runs strong (Bill and Hillary Clinton, who were stung by Kennedy’s 2008 endorsement of Obama, did not attend, but they sent a video that was played at the gala opening Sunday night). Obama, said Kennedy’s widow, Vicki Reggie Kennedy, is a man “my husband loved and admired so much, he gave him a puppy. A man who understands the power and promise of our democracy.”

Bill Delahunt, a former Massachusetts congressman and close friend of Kennedy, said: “It’s affection, a recognition that they both shared something in addition to a political philosophy and policy.”

Jim Manley, a former Kennedy aide, recalled watching Obama slide into a seat next to Kennedy in the back of the Senate as Kennedy walked the freshman through late-night sessions, narrating how each senator was angling to get what they wanted with each maneuver.

Obama and his aides may have idolized Kennedy too much, Manley said.

“I think they hold him out as a model of how the political process is supposed to work — he was a man, after all, that was not afraid to compromise,” Manley said. “But as much as they’d like to hold out Sen. Kennedy’s ideal of getting things done, that’s crushed in the brutal reality which is the House and Senate Republicans.”