President Reagan's smashing victories in Congress this summer in no way end his administration's money battles on Capitol Hill. They are likely to erupt again as soon as Congress returns from its summer recess.

Come September the administration's cuts in spending and taxes will be put to practical tests in the appropriations process, and signs of the big fights are already clear. Moreover, senior administration officials are studying the possibility of using the end of the fiscal year on Sept. 30 as an opportunity to seek new spending cuts that could cause sharp disputes.

Also in September Congress is expected to come to grips with the problems of the Social Security system, as sensitive a subject as ever gets debated in Washington. The partisan contest to win public gratitude for saving Social Security should be intense.

For one fundamental reason, the big fights over the first round of spending cuts and tax reductions cannot be considered final or definitive. Despite the extraordinary success of the Reagan crusade, neither economic conditions nor budgetary realities will permit success for the president's "economic recovery plan" unless additional budget cuts are made soon, and unless national economic conditions improve dramatically. This fact is just beginning to be understood inside the Reagan administration, where internal battles are likely this fall when the need for painful new cuts becomes apparent.

"It's important that we continue the momentum that is building," a senior administration official said last week, referring to budget cuts. "From the policy standpoint it would be wise to strike again while the iron is hot."

And not only for political reasons. The Reagan economic plan calls for additional but still unspecified spending cuts of $29.8 billion plus substantial Social Security savings in the 1983 budget that must be sent to Congress next January. And those cuts won't be adequate to meet administration targets it -- as numerous economists expect -- the administration's optimistic economic forecast for fiscal 1982 proves too rosy.

Indeed, now that Congress has voted to "index" the income tax so that it won't go up in real terms regardless of inflation, the need to continue cutting federal spending could become a permanent fixture of American politics. Only a sustained economic boom or a radical shrinking of federal programs can eliminate that need.

Administration officials will begin to contend with this underlying budgetary reality in the fall, as they prepare the 1983 budget. Informed sources predict an early confrontation over the huge defense program Reagan has supported, since it is one of the few areas where significant cuts could be made relatively easily. Though the president appears committed to the full defense program increasing numbers of key officials around Reagan are worrying about its budgetary implications.

The first congresswional showdowns in September are likely to involve the appropriations bills to finance government operations in the fiscal year beginning Oct. 1.

Since Congress won't return from its summer recess until Sept. 9, there won't be time to enact the appropriations bills by Oct. 1. That means "continuing resolutions" will be needed to keep the government in operation until the appropriations bills are passed.

Theoretically a continuing resolution simply maintains government programs at existing levels, but in fact it can have more significant substantive effects. If the administration had counted on cutting a specific program in 1982, a continuing resolution could force it to spend more money than planned. Or a continuing resolution for a program scheduled to grow -- like defense spending -- could have the effect of holding down spending.

Moreover, Congress has the power to put substantive program changes into a continuing resolution. This is an opening now being studied at the Office of Management and Budget as a possible way to achieve some additional spending cuts beyond those approved this summer by Congress. One possibility would be for President Reagan to threaten to veto continuing resolutions of which he disapproved, a tactic that could allow him again to appeal directly to the country for support while once again circumventing the traditional congressional appropriations process.

The White House has already signaled its displeasure at the apparent willingness of the House and Senate Appropriations committees to stick to traditional congressional accounting methods that could result in actual spending next year higher than administration targets. White House officials have already threatened publicly that the president will veto appropriations bills he considers excessive.

The problem is that the appropriations committees deal in "budget authority," the legal right to spend money. But the OMB is primarily concerned with outlays, the actual spending of government funds. The two are never exactly the same, and OMB Director David A. Stockman now believes that the committees' accounting methods will end up increasing the federal deficit next year.

His publicly expressed concerns have not gone down well with either Rep. Jamie Whitten (D-Miss.) or Sen. Mark O. Hatfield (R-Ore.), chairmen of the House and Senate Appropriations committees. Whitten has accused Stockman of seeking the ability to block specific line items in the budget, an authority the executive branch theoretically doesn't have.

Hatfield expresses a similar concern. In an interview last week he said his committee would perform its constitutional duty, and would not allow the reconciliation process under the Budget Act to dictate appropriations levels. Reconciliation was the legislative vehicle for achieving the budget cuts already approved by Congress. Under reconciliation, Congress can change the level of authrorizations of specific programs; the Appropriations committees cannot vote more money for a program than Congress has authorized.

But Hatfield noted that "we have an ultimate ace" if there is a showdown over the legal standing of the appropriations bills. He said those bills could be written so they begin with the phrase "Not withstanding any other act of this Congress," a legal gambit that could be used to avoid the reconciliation bill. It was a mistake, Hatfield said, to allow important spending decisions to be made through the reconciliation process, in which programs can be profoundly altered "without one day of hearings, without one bit of citizen input," and without serious analysis.

Hatfield quickly added that he wanted to cooperate with the White House, not fight it. But he also indicated that he has an agenda of his own which doesn't precisely coincide with the administration's. For example he said he would like to put off the defense appropriations bill -- by far the biggest -- until all others are acted on. Because of the firm overall spending ceiling set by the budget resolution, defense would then have to be fit into whatever fiscal "space" was left after enacting all the other bills.

Hatfield is an opponent of the sharply expanded defense spending the administration has proposed.

Administration efforts to dominate the appropriations process will meet substantial resistance on Capitol Hill. Just last week, for example, the Senate took up the housing and urban development appropriation bill, after Stockman had publicly indicated his opposition on grounds that it would allow too much actual outlay.Despite Stockman, the Republican-controlled Senate passed the bill 87 to 6.