The unionized half of America's coal industry has been shut down this winter because of two fights - one between miners and mine owners, the other among the miners themselves.

On the surface, the talks to which President Carter has summoned all sides to the White House today will involve only the first and traditional set of issues . . . labor versus management. But underlying this impasse, and complicating it, is the miners' own problem - their union, its leadership and future direction.

Just five years ago, such a problem was virtually inconceivable. Everything was coming up roses for the United Mine Workers.

A reform ticket had taken control of the union and rank-and-file miners were in charge. An energy crisis was brewing; rising coal prices and demand made the future seem solid.

Today, the UMW's rank and file, on the 71st day of a strike that sees no quick end, are talking of impeaching their reform president, Arnold Miller, and staying out until they get what they want.

Over the weekend, the union's bargaining council handed Miller his latest rebuff by rejecting the contract he negotiated with the Bituminous Coal Operators Association, the industry's bargaining arm.

Now President Carter has had to intervene and order both sides to resume negotiations.

Money - even the seemingly attractive 37 percent pay increase that negotiator Miller came up with - seems not to be the central issue. Rather, it is the miners' concern with safety and fringe benefits, and the style and quality of their leadership.

Miller is struggling to keep the UMW together in the face of a fasteroding power base and growing militance among the rank and file, many of them young and restive, over conditions in one of the nation's most dangerous industries.

Miller's problems were clear from the beginning. He and his fellow reformers were elected in 1972 with about 56 percent of the vote. The hefty contract they negotiated in 1974 was ratified by about the same percentage.

That was hardly a strong mandate, and since then Miller's problems have intensified. He split with his reform allies, Mike Trbovich and Harry Patrick. Bright, energetic staff assistants quit in frustration or were fired.

Miller's political eemies accused him of ineptness. His onetime allies criticized him for indecisiveness and railed at his tendency to surround himself with advisers who would not challenge him.

One of those former aides described it this way: "He got rid of everyone who had the audacity to question . . There is no countervailing force on him now to keep him from taking the easiest way out of his problems."

Miller's political problems reached a peak last summer. Patrick, his erstwhile ally, and another rival, Lee Roy Patterson, challenged Miller for the presidency.

The election showed Miller to be a labor general with a very small army.

He was reelected with only about 40 percent of the vote which is to say that three in five miners who voted in the election voted against him. But only about half of the UMW's 260,000-plus members voted; so Miller really was elected by one-fifth of the membership

That weakened base, and the departure from his staff of the lawyers and researchers who helped negotiate the 1974 contract, have left Miller limping.

He hired an outside consultant to replace his researchers. He hired an advertising agency to tell the union's story, in place of the press department he disbanded. The 1974 negotiating team was broken up.

As negotiations got under way last fall and continued into the winter, Miller loyalists complained privately of their exasperation over his frequent disappearances from headquarters and weekend trips to Charleston, W.Va.

Industry negotiators, fully aware of Miller's internal problems, have taken a hard line on the new contract. Gone from the proposed new agreement are features that miners have long taken for granted.

A major objection stems from Miller's acceptance of the BCOA proposal to end the 30-year-old industrywide health and medical care program - heretofore provided at no cost to the miners through a system of royalties on mined coal.

In its place, the industry would put a standard company-by-company health plan, with miners required to pay part of the cost. Coal operators see this as a way to curb the rash of wildcat strikes that depleted the health and pension funds last fall.

The new contract would not grant miners the right to strike over local grievances - a right that Miller insisted he would fight for when he sought reelection.

Until they have that right, miners contend, the crippling wildcat strikes often provoked by local mine conditions, and what they believe is an unworkable greivance system, will continue.

Gone also from Miller's new contract is a right that the UMW did not win until 1974 - the right of a miner to refuse to enter a section of a mine that the miner considered too dangerous.

"These are the things that could lead to dissolution of the UMW," said a former Miller ally. "Job rights are seriously eroded and the UMW has been gelded."

The Miller package featured hefty wage increases that could come with ratification of the contract. However, miners did not put higher wages at the top of their list of priorities at their 1976 convention and again last year in prebargaining conferences. Job security and benefits ranked higher.

"What they want is more control over their lives. It is as though no one is even listening to them," said the former Miller aide.