A woeful lament for the plight of the cities went up from the Conference of Mayors in Louisville last week. But all signs indicate that politically the bestknown big-city leaders are doing brilliantly. How come?

Majority politics seems to be the answer. Most of the mayors have learned how to identify themselves with fiscal responsibility by going against what are widely regarded as givaways to unions, bureaucrats and minority groups.

The troubles of the great cities are not being overstated. Cleveland, Detroit, New York and Chicago have all recently been on the verge of bankruptcy. Schools and transit systems have been temporarily shut down in Philadelphia, Chicago and Cleveland. Crime battens in Miami, New York, Los Angeles and Detroit.

Even more striking, though, is the political success of the big-league mayors. It is not merely that George Voinovich of Cleveland, Jane Byrne of Chicago and William Green of Philadelphia are much more popular than the hapless figures who were their immediate predecessors. Edward Koch of New York has just been endorsed for reelection by both Democratic and Republican parties. Tom Bradley of Los Angeles won reelection without a runnoff and is a fair bet to be the next governor of California.

A tough stance on financial questions is one obvious reason for the high standing of the mayors. Green, Voinovich and Byrne have been squeezing hundreds of millions out of city budgets. Koch has achieved -- on paper, at least -- surplus. Compared with those who went before, those now at city hall come on as scrooges.

In making economies, most have had the skill to take on groups that are little loved. Labor, first of all. Byrne took a three-day strike in order to cut down on cost-of-living adjustments to transit workers in Chicago. Bradley won a public referendum against increases in police and fire pensions. Green had pay increases in Philadelphia to zero during the past year. Koch counts as his chief enemies in the Big Apple the labor lawyer Theodore Kheel and the head of the municipal unions, Victor Gotbaum.

Bureauctratic structures provide a second target of opportunity. Most major cities parcel out power among a network of independent authorities. Courts and judges also have a say. These institutions regularly draw the fire of successful mayors. Judges and the transit authority are favorite punching bags for Koch. Green has publicly invited members of the Philadelphia school board to quit so he can appoint his own people. Byrne won a victory over Gov. Jim Thompson of Illinois by asserting influence inside the municipal transit authority. Even Kevin White, the much harassed, veteran mayor of Boston, has been able to score points at the expense of the school board there.

Lastly there are the minority groups receiving assistance in the big cities -- especially, at least in the North, the blacks. The subject is a delicate one, and I should emphasize that what follows is strongly impressionistic. Still my impression is that in the major cities, mayors skillful in using the local idiom -- at home in the native style, so to speak -- can win approval by showing that minorities do not receive special favors.

Koch came out of the pack in New York by asserting himself -- in opposition to most black figures -- as being for the death penalty. Since then, in the feisty way dear to New Yorkers, he has said of blacks the kind of things that has earned him the sobriquet "urban redneck."

Bradley, a black himself, personifies the more "laid-back" style of the West Coast. It is notable that he did not lift a finger to stop the action by white suburbanites that brought the end of busing in Los Angeles.

Byrne specializes in dramatic assertion of personal authority that suits the Chicago style. Her move to a public housing project expresses the community's belief in equal access to municipal services -- but without making any serious change in the status quo.

If this view is correct, the solution of urban problems -- the improvement of schools, transit and personal security -- does not follow what the best-known mayors are doing. Rather, the actions of municipal leaders testify to the nature of present-day politics. Increasingly beset by incomprehensible problems and more and more split from party affiliations and separated from each other by the privatizing force of television, Americans project their hopes and frenzies on individual leaders. In the great cities the mayors are urban champions doing combat for ordinary people against the system.