In California last week, the Thiokol Co. finished a "burst test" on the first-stage engine of the new MX intercontinental ballistics missile to see how much pressure it could take before the casing split.

Thiokol, along with Aerojet General Corp., the Hercules Co., and Rockwell Intgernational, the contractors that have built prototypes for the other three stages of the giant MX, has been working on these engines for more than three years.

At the Pentagon this week, officials were working out a plan to lower the yield of each of the 10 warheads the missile is to carry in order to save valuable nuclear materials that could be used in order new weapons.

While President Reagan and Defensef Secretary Caspar W. Weinberger ponder the future of the MX and its controversial land-based shelters, final development work on it that began in earnest over three years and $3 billion ago continues unabated.

Developing, designing and building a new strategic missile, whether land or submarine based, is a 10-year process, according to Pentagon and defense industry officials. The MX is halfway down that road with the first 10 operational missiles scheduled to be ready in Decembedr, 1986.

Reorts that Weinberger is recommending to the president that the MX be scrapped in favor of a socalled "common missile" that would be used by the Air Force and the Navy has top civilian and military brass in those services along with defense specialists on Capitol Hill shaking their heads.

"It would be the beginning of another 10-year process," one official said yesterday.

The first Air Force contract for the MX, according to one industry source working in that program, was an advanced development program awarded to Aerojet in May, 1974.

Other contracts followed as first the Ford and then Carter administrations attempted to determine the specific type of missile they wanted. Contracts for developing the enginesx for each of the first three stages were awarded in 1978, but it wasn't until October, 1979, that President Carter, after turning down the common missile idea, selected the large, 91-inch diameter MX, which is too large to fit into a Trident submarine.

Despite Reagan's publicly voiced doubts about how to base the MX, work on the new ICBM has continued during the first seven months of the Reagan administration.

For example, in New Mexico earlier this month the proposed inertial guidance system -- which would feed the MX's position and velocity to the flight computer after launch -- was subjected to a rocket sled test to see how it performed under stress. The sled got to a speed of about 1,000 miles per hour then was quickly stopped in an attempt to imitate the thrust of a launched rocket and the later effect of traveling in space.

At Vandenberg Air Force Base in California, work is under way on the $50 million complex of assembly buildings and launch pads that would handle the first five test firings of the MX, scheduled to begin in January, 1983. t

In New York City a telecommunications consulting company and an architectural engineering firm are preparing reports on how to build a new town of offices and homes for the 18,000 people that would operate the MX system.

In Las Vegas, a civil engineering concern has just put the finishing touches on a study of regulatory constraints and drainage problems associated with building such a new MX base town in Nevada.

At the Nevada nuclear test site last Tuesday, an Air Force contractor put the finishing touches on five precast concrete sections that, when put together, become the horizontal shelter for the MX, part of the controversial, Carter administration shell game, land-basing scheme that forced the Reagan administration review of the entire MX program.

Air Force officers supervising the shelter program have not even made changes in the Carter approach that they know the Reagan people want. One is removal of the so-called verification holes built into the shelters, cement blocks that could be removed in the roof to allow Soviet satellites to confirm how many MX missiles the United States deployed. Reagan wants them removed to save money and because he isn't interested in helping the Russians with verification.

"We could save money by taking that out," an officer said yesterday, "but we haven't done it."

Testing of the precast horizontal shelter and another version, built by a specially constructed cement-pouring machine, will continue at the Nevada test site over the next few weeks despite the up-and-down rumors that the Carger program has been shelved.

"The technology for mass producing shelters," the officer said, "could be used in another shelter program," though he quickly added that close to a year could be lost if the president chooses a land-basing scheme that needs vertical rather than horizontal shelters for the MX.

Under the Carter program for the coming fiscal 1982 year, $2.4 billion was allocated to development of the MX, a figure that Weinbeger heartily supported in appearances in March before House and Senate committees.

"We have no doubt whatsoever that the MX missile is needed," he told the Senate Armed Services Committee on March 4, "and we are proceeding on that basis . . . . The question is where to put it, how to base it."

Regt basis . . . . The question is where to put it, how to base it."

Reagan's proposed fiscal 1983 budget contains an additional $2.3 billion for MX development along with $1.8 billion to begin procurement of the first missiles and another $1.8 billion for initial basing.

The current schedule, drawn up by the Carter Pentagon, called for initial flight testing of the MX beginning in January, 1983. Then in July, 1983, the final production decision by the president was to be made, so that the first missiles would be operational by the end of 1986.

"We are up against that schedule right now," an industry source said yesterday, "and any change in the program will surely push that date back."