Canadians have had a love-hate relationship with Prime Minister Pierre Trudeau since he burst onto the national political scene 14 years ago. But his popularity rarely has been lower than now, with three-fourths of the people saying the Trudeau government has dramatically mishandled the economy.

Trudeau's nationwide overall approval rating of 28 percent is a distant third behind Conservative Party leader Joe Clark's 40 percent and New Democratic Party leader Ed Broadbent's 43 percent, according to the Gallup Poll.

Even in Trudeau's native Quebec, only 45 percent of the population approves of his performance, and that figure plummets in the western provinces.

With Canada facing its worst economic crisis in 30 years and embroiled in regional and ethnic rivalries, observers consider Trudeau's Liberal government unlikely to make it to scheduled elections two years from now.

Canada's Conservatives say the Trudeau government has become too interventionist, while the leftist New Democratic Party thinks the government has done too little and ought to be looking to more aggressive public investment programs, broad industrial development policies and tax cuts to stimulate consumer spending.

Clark, who held the prime minister's post for less than a year, beginning in 1979, accuses the government of "economic deception." Conservative House of Commons spokesman Michael Wilson called the Liberals' handling of the economy "a mosaic of failure" and said, "It is a political strategy designed to be seen as doing something."

"We are at war with the Trudeau government," Ronald W. Lang, director of research for the Canadian Labor Congress, said after the government introduced measures to hold federal employes' wage increases to below the rise in the cost of living.

Managing the economy is but the latest and most immediate of the problems that make governing Canada difficult. The nation is deeply divided along regional lines. Even the adoption of a national constitution last spring, the culmination of Trudeau's efforts to meld Canada's heritage into a single national identity, has not brought Canada together.

To the east are the continuing problems of French-speaking Quebec, the base of Trudeau's Liberal Party support but simultaneously a hotbed of antagonism toward the federal government and the rest of the country's English-speaking provinces.

Another brand of militant opposition to Ottawa comes from the west, where the leadership of provinces with abundant energy resources, particularly Alberta, is increasingly hostile to the Trudeau government. Alberta's provincial prime minister, Peter Lougheed, regularly criticizes what he calls Ottawa's "socialists" and said recently that the Liberals' "antiprivate-sector investment policies" and accompanying high interst rates are "causing serious alienation" in the west.

On June 28, the government brought to Parliament a new budget that estimated next year's federal deficit at nearly $20 billion--twice the amount forecast by the government in November 1981. The revised figures startled the public, although economists and other close observers of Canada's financial fortunes had been suggesting all year that the deficit projections were short of the mark.

"It was clearly apparent by November that our whole economy was turning down much more than they realized," said J. Pearce Bunting, president of the Toronto Stock Exchange.

Organized labor, which is more militant here than in the United States, has focused its criticism on the governments campaign against inflation. "We reject their preoccupation with inflation," said Robert White, director of Canada's branch of the United Auto Workers.

Big Three auto contracts expire in mid-September, and both White and auto industry officials are predicting that a strike will follow. Ford executives have threatened to pull the industry out of Canada if the unions do not moderate their demands.

In an effort to halve the nation's inflation rate of about 12 percent, the government announced a program to limit civil service wage increases for the 500,000 federal employes to 6 percent this year and 5 percent next year. Last year the federal employes received 12.2 percent raises. The program was expanded last week to include workers for private railroads, and the federal government encouraged similar wage restraint by the provincial governments.

The new budget also called for eased rules on the government's monitoring of foreign investment, a slowdown in the campaign to increase Canadian control over the country's energy resources and a new jobs program. It included a housing subsidy program to provide $3,000 grants to buyers of new homes and to people buying their first homes.

When the new budget was proposed, Finance Minister Allan MacEachen called on Canadians to build a "tougher, more resilient" society.

"The way out of the recession is to bring down inflation and increase productivity," he said. "We must all share the burden and the responsibility."

Business in Canada, sensitive to its dominance by outside ownership, is only beginning to speak up in Canadian life. The business leaders' not surprisingly have criticized the National Energy Program, which aims at increasing Canadian control of energy resources, and other efforts by government to tighten its hold on the economy.

Among these many critics there is general agreement that the recession is in large measure the product of both the worldwide recession and the U.S. monetary policy, which has produced high interest rates in the United States and has been a key factor in Canada's high rates as well.

MacEachen conceded that the government "cannot blame the rest of the world for our inflation rates," but he firmly pinned the broader responsibility for the nation's weak economy on outside forces.

"What is happening in Canada is basically the manifestation of the malady that is gripping all the industrialized countries," MacEachen said recently.

In November, the Liberals said they expected the economy to grow at a 3 percent annual rate. By June 28 that estimate was revised to show a decline of 2 percent, costing the government $4.5 billion in lost revenue.

What happened was that the government did not anticipate either the drop in world energy prices or the continuing high interest rates. Those two factors cost the government budget close to $3 billion. The National Energy Program delayed the impact of the oil price increases of the mid 1970s, but at the same time frightened some foreign investors.

"The program met what the government thought was a very real need--they needed a lot of money to fund their deficits," said James Livingstone, president of Imperial Oil Ltd. Three quarters of Imperial is owned by Exxon.

As a result, however, he said, "the petroleum industry has had significantly less money than expected." Imperial paid about $91 million in unanticipated taxes in 1981 and expects to pay $150 million in the new energy taxes this year.

Recently, the company virtually abandoned plans for its mammoth $13 billion Cold Lake Project, which by the end of the decade was to have produced about 10 percent of Canada's crude oil requirements, about 140,000 barrels of synthetic crude a day.