In the privacy of the White House family quarters, President Carter has begun to campaing for reelection in the low-keyed, personal style that was so successful for him four years ago in living rooms from Iowa to New Hempshire.

Last week, the president held the first in what his aides say will be a regular series of meetings with small group of people from whom he will be seeking advice and, untimately, support.

In some ways, the meetings are a continuation of the process Carter began with his Camp David "domestic summit conference," where he consulted with more than 120 people summoned to the presidental retreat to discuss energy and other problems besetting the administration.

Yesterday Carter hosted a larger lunch for the people he consulted at Camp David. Eighty-two people, more than a third of them government appointees, showed up to hear the president thank them for their advice and ask for their continued help "in the energy field and the attitudes of out nation."

Carter also touched on the purge of the Cabinet that followed the Camp David discussions.

"i have made some good decisions since we left Camp David,"" he said. "some have been controversal. But I think they have been sound. There is no doubt in his mind that I now have a stronger and more cohesive and perhaps even a more competent Cabinet. But at least we will be stronger and more cohesive . . . . We have replaced excellent people with excellent people."

The smaller gatherings the president now plans, featuring what one participant called a good deal of "social chitchat" amid the trappings of the White House, appear to be a reaching out to potential sources of support for the reelection campaign.

The first group Carter met with include Gov. Julian Carroll of Kentucky; Mayors Kenneth Gibson of Newark, Earnest (Dutch) Morial of New Orleans and Richard Caliguiri of Pittsburgh; J.C. Turner, president of the operating engineers union, former Virginia lieutenant governor Henry Howell and former Maryland Senator Joseph D Tydings, who is now a Washington Lawyer.

This first group, accompanied by wives, was met personally at the diplomatic entrance to the White House by the president, who excorted them to the upstairs family dinning room. Lunch, according to one participant, was a "friendly, homey get-together." When it was over and Carter had to leave for a meeting, his wife, Rosalynn, took the group on a rare tour of the family quarters.

One participant noted that the first group consisted largely of early supporters of the president from the 1976 campaign or people who generally had supported the administration since the election.

Making the connection with the 1980 campaign, he said, "I think the president is making a effort to gather his early friends and supporters around him."

A source fimiliar with the meeting said, "It looks to me like it's Iowa politics in the White House, which apparently is what he's good at."

In the view of his aides, Carter is at his best with small groups such as the one invited to the White House last week and those that will follow.A virtually unknown candidate in 1976, he parlayed tireless, personal campaigning in homes and anywhere else people would gather into surprising victories in the early political test states of Iowa and New Hampshire. From that foundation, he built the momentum that eventually captured the Democratic presidential nomination.

Now the president appears to be embarking on the same course, with the added advantage that he can draw up the guest list and host the gatherings in the impressive setting of the White House.

Confirming that more such meetings are planned, presidential assistant Jack Watson said in an interview that the effort is 'a continuation of the Camp David process."

"he [Carter] wants to do it on a regular basis, to consult with people," he said. "the object is to get dirrerent perspectives that represent different points of view, different parts of the country, a cross section of political leaders and private sector leaders."

Watson denied that the meetings are an attempt to build a network of friends for the reelection campaign, but conceded that with 1980 approaching, politics is an increasingly important factor in the White House.

"that the 1980 campaign is not going to be the rationale," he said. "But to be sure, friends and supporters are the kind of folks he wants to consult with." CAPTION: Picture, Carter at White House lunch yesterday; in foreground, Energy Secretary James Schlesinger, AP