Cameras could capture moments. "Shivering people wrapped in blankets singing "Amazing Grace" at the Lincoln Memorial. The woman auto worker who camped out on the Capitol grounds at 4 a.m. to get a good place for the swearing-in. The proud high school bands that strutted along Pennsylvania Avenue. The ragamuffin Peanut Brigade of people - plain people - in the inaugural parade. And the family of Carters who walked down Pennsylvania Avenue - father, mother, daughter, sons and grandson - capturing the crowd and the spirit of inauguration day.

Mary Harris, a Washington resident, who has seen many such occasions, said it as well as anyone. "This is the most beautiful experience I've ever had in my life," she declared, as Carter stroke by. "I just wanted to reach out and put my arms around him. Can you imagine him walking? I just wanted to let him know that he didn't have to worry about doing the job all by himself, that he could certainly count on me."

If each inauguration has its own rhythm, its own sense of pace, yesterday's was one of an optimistic return to normalcy, a reaffirmation of hope - a hope with a distinctively Southern flavor. The old outsiders were coming in, and they had Southern accents.

Nothing symbolized this more than Carter's walk from Capitol Hill down Constitution and Pennsylvania Avenues to the White House. "He's walking, he's walking. Jimmy is walking," people shouted all along the parade route. "You wouldn't see no Yankee President doing that," said one man, obviously a Southerner, as he stood on the sidewalk near the Taft Memorial.

It was neither the biggest nor the smallest inauguration in recent times. Officials estimated the crowd along the parade route at 350,000, that at the swearing-in ceremony at 150,000. This was larger than 1973, or 1969, when the crowds were estimated at 300,000 and 200,000 persons respectively. But it was far smaller than 1965 when an estimated 1.2 million viewed Lyndon B. Johnson's inauguration.

The Carters approached the White House three abreast - Amy between her parents, holding each by the hand. She left them at the reviewing stand, as they continued down the parade route to 17th Street and then turned around and walked back. With perfect timing, they met Vice President Walter Mondale and his family as they reached the White House, also now on foot after leaving their car.

That meeting was typical of a day of events that, in general, went off with similar precision. There were no major delays or traffic jams, events started on time and ended more or less as predicted, and primary discomfort for the crowd was the cold.

The people who came were here to celebrate in their own way a man they expect to renew their faith in their government and their way of life. Prayer and Song

Some 5,000 of them came to the Lincoln Memorial in the cold early morning to pray and sing and hear Martin Luther King Sr.'s powerful voice tell them of the poor and hungry who must be fed, the love that must be given to God, and his hope for the new President.

"God grant that our President will remember always the least of us, 'cause there always will be so far as I know, more of the least of us than all the rest."

As King spoke, choruses of "Amen." and "Yes sir," and "Praise the Lord" came spontaneously from the crowd.

As always during this day, reminders of real life were never far removed from the fun and the ceremony. As King, the Rev. Bruce Edwards of Plains Baptist Church, and Carter's sister Jean Stapleton spoke, airplanes from National Airport drowned out their words for precious moments of the 35-minute service.

"I came to pray for this man," said Washington Auxilliary Biship Eugene Marino, one of the Clergymen who attended the prayer service. "He needs all the prayers he can get."

They wore fur coats and parkas, knitted caps of all colors, mufflers and scarves and anything else to keep out the cold. The sang firmly, aided by a 400-voice choir and opera singers Leontyne Price and Sherill Milnes.

"We came here to praise the Lord for Jimmy Carter," said Ilene Flerner, a nurse from Phoenix, Ariz. The Troops Were Gone

In a city so often torn by tension the last decade, security was markedly relaxed. Protest demonstrations were poorly attended. No one threw garbage at the President when he passed as they did in 1973 at Richard M. Nixon's second inauguration. Hardly anyone even thought of it.

When Carter thanked his predecessor, Gerald Ford, the accidental President, for "healing the nation," city official George S. Rodericks watching television at Mayor Washington's command post, said, "I remember four years ago, the last inauguration, out in the street . . . the rioting . . . the hate."

When Carter opened his speech by thanking Ford, the crowd at the Capitol applauded and cheered so long for the outgoing President that he had to stand up twice. Ford appeared close to tears.

"Some attitudes have changed," Rodericks said later. "Two or three years ago, something like Carter walking was unthinkable. For Nixon's inauguration they lined up troops every 10 feet. They called it an horror guard. But the truth of it was it wasn't."

By 4 p.m., the only incidents reported to police were the arrest of three pickpockets and a few street vendors for operating without a license. The only injury reported occurred when a woman identified only as Jeanne Braceland, 50 bruised her nose when she fell from a horse she was riding sidesaddle in the parade. One young man was arrested for the possession of a bag of amrijunana.

Everywhere there were crowds. Everywhere there were Jimmy Carter buttons. Everywhere they were chilly bodies, braced against the worst cold spell in the city's memory. Everywhere there were scenes of people straining to catch a glimpse of history.

"I really wanted to hear what Carter had to say," said Charles Newell, a 17-year-old senior at St. Albans School, who climbed a tree limb on the Capitol grounds to watch the swearing in. "I think people are having a good time. It's like a big celebration."

Newell, like many at the parade, lives here. It was the out of towners, however, who gave the day its flavor. They were the Carterites, the old friends, the Georgian neighbors, the campaign workers who had come by bus, plane, train and car from across the land.

Yesterday was their day. They were the people of hope, the ones who talked of changed and cerebration.

"I had chills inside of me," said Cornnie Freniere, one of the faithful, who had come from her hometown of Fairborn, Ohio. "He symbolized the people, the real people, not the ones in the top hats who stay inside their cars, but the real of people of America. I was stunned when I saw his beautiful face and beautiful smile. To see him leading the parade like that was everything America stands for."

Ironically, the bitter cold that so many had felt would spoil the day, turned out to be a common bond. It affected the rich as well as the poor, Northerners as well as Southerners, blacks as well as whites.

Billy Johnson, his wife Connie, and son Billy, bought neck to ankle, fiber-filled winter suits to wear at the parade before they left Americus, Ga. "We were ready for rain, sleet or snow," Mrs. Johnson said. "We weren't going to miss this for anything. But we've been surprised. It was at least this cold when we left Georgia yesterday. Seeing the Swearing-In

At the swearing-in in front of the Capitol, the stars of the government establishment performed their rituals on a stand high above the people. Those who did not have tickets for the ceremonies milled around the Supreme Court building and listened to the oaths and the speech on transistor radios and, with difficulty, over loudspeakers.

The senators and congressmen - all but a handful white and male - filled either side of the platform. Below them, the all black Atlanta University Center Chorus stood, black-robed, ready to sing the Battle Humm of the Republic. Their rendition was one of the highlights of the ceremony.

From a distance, cheers could occasionally be heard from the people watching and hearing their leaders.

"It's like going to a rock concert," said Gary Holland of Columbus, Ohio, one of those who had a ticket a attend the ceremony. "You sit in row 'I' or something and can't see anything but you hear everything and at least you can say that you were there."

Connections were not always the key to getting in to the swearing-in ceremony. A general in a cashmere coat shoved up against a shoe salesman in a double knit raincoat in one of the groups of people trying vainly to get into the Capitol grounds.

"But I've got a ticket," someone yelled. "Everybody's got a ticket," someone yelled back.

"I guess one of the things about this republic is that there are 200 million people who really believe they have an absolute right to sit next to the President," said Edward J. Nevin III, a San Francisco lawyer who was also trying to get in.

At times there was tension between the masses and the police assigned to keep them out and orderly. Some people did not realize they needed a ticket to get in, and others who had tickets couldn't get past the others who didn't.

Then when the ceremony was almost over, people who wanted to leave were prevented from doing so by the police. People began pushing and shoving and one little boy who was pushed toward a policeman was bounced fiercely back into line by the policeman's stomach. A Crowd-Pleasing Parade

The parade, with its almost endless high school marching bands and state floats was the largest, best attended event of the day. Inevitably, there were those who didn't like it, or couldn't see. "Parades are for people who are more than 5 foot six feet tall," complained Patsy Wells of Plains, Ga. "Here I came all the way from Massachusetts and I can't see a thing," said Walter Olszewski, of Ware, Mass. "I can't see anything and I've lost my wife and granddaughter."

But the overwhelming mood of the crowd, which varied from one to 10 deep along the parade route, was relaxed, and most everyone seemed to enjoy the spectacle.

A few climbed trees or stood atop portable toilets, Army jeeps or school buses. Some brought ladders from home to assure themselves a view."I found out from the July 4th parade that you've got to have one of these ladders to be sure to see," said John Sadler, of Arlington, Va.

Some persons had been waiting two hours in the 30-degree temperature before the parade began in the shadow of the Capitol. They huddled together, slipped coffee, jumped up and down.

When the military units and bands started moving forward, they edged foreward, hoping to catch a glimpse of the new president in a limousine. The supense built as more than a dozen trucks and cars carrying television cameras and newsmen passed by. "There's Harry Reasonable, he's my favorite," shrieked one young woman as ABC newsman Harry Reasoner passed. "Where's Barbara, where's Barbara?"

A hush swept through the strong when Carter suddenly appeared walking. "I don't believe it. I don't believe it. Jimmy is walking right down the middle of the street," grasped Mary Lou Johnson os Siouz City, Iowa, who waved a huge paper-mache peanut above her head. "He's out there walking."

And, indeed the new President was. There was Jimmy Carter walking down Pennsylvania Avenue, proud, relaxed confident.

The Secret Service had been alerted weeks ago that Carter intended to walk, but the dozens of agents around him looked shocked. The people loved it.

"Man, he got out and walked among the people like a man - not like Nixon," said Jasper Lewis, a black man from this city. "He has got no reason to be scared."

The parade itself was a typically American, electric collection of floats, stunts, march steps and people. There were many young women in skimpy outfits and even more men wearing uniforms with long pants. The luckiest man in the whole parade was probably the man portraying black explorer Matthew Alexander Henson on the Maryland State Float - his costume was a fur parka and everything else one would wear while exploring the North Pole.

The Carter and Mondale families appeared to enjoy the whole 1 hour and 45 minutes of the parade. Carter's grandson, Jason, who is nearly a year and a half old, toddled around the front of the reviewing stand, cocoasionally being cuddled by his aunt Amy, having his nose wiped by his grandmother Rosalynn, and being dawled on the knees of his frandfather the President. The Mondale's youngest child, wearing a hooded parka, sat on the floor with Amy and Jason most of the time.

Hucksters lined the route. It appeared tha Jimmy Carter may be the most marketable Presidnet in history. Yesterday the peddlers had Jimmy Carter buttons, Jimmy Carter banners, Jimmy Carter T-Shirts. Jimmy Carter stocking caps. Peanut key chains sold for $1, peanut necklaces for $2, and a platter with Carter's ware house seal on it was going for $20.

Sales, however, went slowly. By midafternoon, John Nelson, of Norfolk, Va., said he had sold only 450 of his allotment of 700 Carter peanuts.

"By 6 o'clock I hope to be doing a big business," Joseph Shelton, who sold stocking caps at 15th Street, said hopefully. "It's getting colder now and the people have been standing up for a long time."