In 1976, General Dynamics proposed building a new, shoulder-fired antitank rocket for the Army that it said would cost only $78, half the price of the short-range rocket launcher then being bought from Norway.

Today, after years of development and testing, the Army wants to buy 700,000 of these rockets, called the Viper, but the price is now nearly $1,000 each.

The Viper is only one of many new Army weapons that has escalated in cost and failed to meet original performance standards. As a result of its attempt to rush into modern weaponry after Vietnam, the Army is facing an unprecedented number of cost overruns on major new systems, according to administration and Capitol Hill sources.

And at a time when Congress is looking for places to cut defense spending, the Army budget, according to sources on Capitol Hill, may represent this year's prime target.

"In its rush to replace block obsolescence from the 1960s," one administration official said of the Army, "it could not manage all those programs at the same time."

This week, the Pentagon is expected to send Congress a newly required report on weapons that within the past few months have exceeded their previously expected unit cost by more than 15 percent. At least five Army systems are involved, sources said.

Among them:

The Patriot air defense system. Its acquisition cost has grown from an estimated $5.5 billion in fiscal 1980 to $7.8 billion in fiscal 1982. Another increase is forecast. It, too, has been having technical problems.

The AH64 Apache helicopter. Its cost has jumped to an unprecedented $16 million each. Just two years ago the price was put at $10 million per helicopter. The Army and Hughes Helicopter, the builder, are still trying to come to an agreement on price.

The 108 Pershing II missiles destined for deployment in Europe. Their cost has gone to nearly $2 billion, almost double the $1.1 billion projected last June, according to Capitol Hill sources. The cause of the increased costs, sources said, was a decision to slow down production and stretch delivery of the new missiles by three years.

But several other systems have had cost and technical problems.

The Copperhead is a laser-guided 155 mm artillery shell. For fiscal 1980, the Army proposed buying 110,236 at a cost of $1.2 billion. By fiscal 1981, only 44,386 Copperheads were scheduled for purchase. But the total price held steady at $1.1 billion.

The Army could not increase production this year, and thus gain some cost reduction on volume, because the shells could not demonstrate the required reliability during test firings. As a result, the Marines dropped plans to purchase the Copperhead in fiscal 1982.

The costs for new Army armored personnel carriers, called fighting vehicle systems or FVS, are also skyrocketing. Between December, 1979, and December, 1980, the acquisition cost rose from $7.4 billion to $13.1 billion. Each vehicle is now expected to cost upwards of $1 million. Inflation is blamed as a major factor, but Pentagon officials told Congress that the builder, FMC Corp., underestimated costs.

The Viper program illustrates some of the weaknesses in the Army's approach on big weapons.

At the outset, GD won a contract to design the system at a $78 per unit cost when the Army's current shoulder-fired antitank weapon, the LAW, cost about $135 apiece.

The Viper was not only going to be cheaper, but it was going to be easier to fire and have a longer range. It was to be a self-contained system, with the fiberglass carrying case serving as the firing tube. Once the weapon was fired, the case was to be discarded. In addition, it was to be effective out to 200 meters where the LAW was considered good for only 100 meters.

But a December, 1980, review found several problems with the Viper, including duds and a noise problem. When the Viper was fired, it apparently created a sound that was considered dangerous to the soldier's ear.

The contractor attempted to solve the noise problem by lengthening the tube. In March, 1981, when the new version was tested, five inches of the tube blew off.

More tests were conducted last year. The General Accounting Office, after reviewing them, recommended the system not be permitted to go into production.

The Army, however, thought differently and began negotiating a production contract with GD.

Meanwhile, Congress started asking questions.

When it made a cost-effectiveness comparison, the Army acknowledged that at ranges up to 170 meters, the LAW was cheaper as a tank killer. Thereafter the Army began to emphasize the need for a system that could destroy targets from 250 to 350 meters away, far beyond the LAW's range.

The Viper failed to meet a water emersion test of two hours. At most, five minutes was the longest it could be left in water. The Army responded that such a test had to be re-evaluated and that the system had passed a rain test.

The one test the Viper has still to pass, however, is cost. In approving its production, Undersecretary of Defense Richard D. DeLauer wrote that the Viper's price should not be "greater than about double the cost of the current . . . LAW."

The LAW's current cost, according to the Pentagon, is about $250. Now, not even the Army thinks the Viper can be produced for less than $500.