Can Schmidt hang on in West Germany?

Is West Germany slowly becoming ungovernable?

Two years ago, when Chancellor Helmut Schmidt's left-liberal government was reelected with a solid majority, the question would have appeared absurd. But after the perplexing result of Sunday's state election in the chancellor's home city of Hamburg, many West Germans are starting to wonder if his government can survive.

The vote was a clear defeat for Mr. Schmidt. Despite running a campaign pinned to his personal appeal, with the slogan ''Hamburg won't leave Helmut in the lurch,'' his Social Democratic Party (SPD) crashed from 51.5 percent of the vote to 42.8 percent--its worst-ever result in the city-state.

Mr. Schmidt's Bonn coalition partners, the liberal Free Democrats (FDP), failed even to win the 5 percent necessary to take seats in the city parliament. They scored 4.8 percent.

The conservative Christian Democrats (CDU) gained more than 5 percent to finish with 43.2 percent--a major advance but nowhere near enough to form a government.

The biggest shock of the election was the success of a new third party, the ''green'' Alternative List, which entered the city parliament for the first time with 7.7 percent, although it had rejected in advance a coalition with either major party.

It was the second straight success for ecologist-leftist groups in regional polls. They won entry to the Lower Saxony assembly in March and are already represented in West Berlin, Bremen, and Baden-Wurttemberg.

Their election themes are appealingly simple--''no'' to nuclear energy, ''no'' to new US missiles in West Germany, ''no'' to pollution of the environment. Their voters are overwhelmingly young, and their brand of militant politics is a challenge to the whole political system.

If the Hamburg Greens' success were repeated in a federal election, it would quite possibly mean that no ''establishment'' party or coalition could secure an overall majority.

The Greens underlined their image as the ''naughty children'' of this affluent society on election night. They were unceremoniously cut off television when one of their candidates attacked President Reagan's visit to Bonn this week.

For Mr. Schmidt's shaky Bonn government, the result could hardly have been worse. The poll showed that the SPD is continuing to lose voters in droves both to the Greens on its left and the CDU on its right.

But it also showed that Foreign Minister Hans-Dietrich Genscher's FDP is being dragged down with the Social Democrats, losing its cherished role as the natural coalition partner for both major parties.

In Hamburg, where the Free Democrats committed themselves in advance to a coalition with the SPD, they failed to clear the 5 percent hurdle. Under the West German Constitution, parties that score less than 5 percent get no seats at all. Many on the right of the party will draw the lesson that only a coalition with the CDU can save them from extinction.

Mr. Genscher was cautious on election night. ''We are all somewhat helpless, '' he told an interviewer, adding that theresult showed ''we must make our own positions as a party clearer to the voters.''

That could signal tougher bargaining by the FDP in next month's government negotiations over the 1983 budget. It could also mean that the liberals in the state of Hesse will come out in favor of a coalition with the Christian Democrats before the state votes in September.

No national FDP leader is yet speaking publicly of deserting the Social Democrats in Bonn. But if the Hamburg pattern is repeated in Hesse, pressure for a change, already strong in the press, could grow.

The result was only a mixed blessing for the CDU. Opinion polls show that if federal elections were held tomorrow, the party would probably just win an absolute majority.

But the next scheduled national poll is not until 1984, and by then the advance of the Greens could mean that the CDU too is not in a position to form a majority government.