However much the old, wartime "special relationship" between the United States and Britain may have eroded over the postwar years, it is alive and well in the rapport between British Prime Minister Margaret Thatcher and President Reagan. They came to power at roughly the same time with roughly the same ideology. They like, and complement, each other.

Before Reagan was a stunningly successful politician, he was a movie actor, and it shows in his episodic approach to the presidency. Before Thatcher was equally successful as a politician, she was a schoolmarm. And that shows, too, in the way she does her homework and her tough-minded, classroom manner and command.

All this is by way of setting the stage for her arrival here tomorrow. According to U.S. officials, she passed the word that she wants to get down to the nitty- gritty of arms controls, among other things, in the quiet informal confines of the president's Camp David retreat. She will be bringing not just Britain's view and, by extension, Western Europe's on arms-control prospects, but also an exclusive, firsthand insight on the Soviet state of mind derived from her recent talks with Mikhail Gorbachev, said to be No. 2 in the Soviet power structure.

Their encounter could have more effect on U.S. arms control policy than the president's sessions with his own arms controllers. The effect would be to strengthen Reagan's own influence on his sharply divided administration's effort to work out a U.S. position for the crucial meeting between Secretary of State George Shultz and Soviet Foreign Minister Andrei Gromyko.

Of all the allied leaders, "she is the only one who can lean on him," says a State Department oficial. But Thatcher will be careful not to lean in a way that would threaten the desired effect: the removal of "Star Wars" as a hobgoblin for European allies and a stumbling block in arms negotiations with the Soviets.

Star Wars -- formally known as the Strategic Defense Initiative (SDI) -- got its misnomer by the loose way it was introduced by the president in his speech on March 23, 1983. To hear him tell it then, and later in the second presidential debate, SDI would eventually make nuclear weapons "impotent and obsolete."

Critics do not quarrel with the value of continuing to explore the possibility. What they deplore is the excessive hype and the danger that this will merely provoke an incredibly costly and destabilizing escalation of the arms race, dooming serious efforts at arms control.

That, I'm told, will be Thatcher's measured argument. She will tell the president that, like it or not, defensive weapons are now inextricably caught up in the arms-control process; that the issue will have to be met before the Soviets will proceed on other fronts (the control of strategic and intermediate- range offensive missiles) and that this requires a serious effort to halt the extension of the arms race into space.

Thatcher will not be unwise enough to insist that Reagan abandon his dream -- merely that he put it in some realistic perspective by getting it, insofar as possible, back into the closet of research and development. That's something that cannot be controlled by negotiated agreements; controls on R&D cannot be effectively verified. Mutual restraint must center on deployments.

Thatcher will be bringing with her the impression from her talks with Gorbachev that the Soviet Union can't afford a defensive nuclear arms race, that it would rather put the money to better economic purposes, but that it will not hesitate to try to match American technology. With no curbs on ultimate deployment, the effect on WesteEurope would be to inflame latent fears that the United States will never be willing to risk its cities to save Europe's -- the essence of nuclear deterrence as the Europeans see it. The British and the French would take no comfort from a developing U.S. nuclear defense that would rob their own forces of the desired deterrent effect.

"Star Wars" in its worst, most idealistic and unrealistic formulation is widely seen, even in most administration quarters, as a genuine threat to arms control. Thatcher is uniquely positioned at the moment to make the case. The only question is whether the president, with his public commitment to the notion that there is some kind of shortcut to a nuclear-free world, is of a mind to make the best use of her second opinion within his own divided counsels.