This is not all-out war.

The American and British attack on Afghanistan is confined to a strictly limited selection of targets.

The military objectives are closely related to the delicate diplomacy that been going on since 11 September.

The US government is balancing a series of competing demands.

Firstly, it wants to punish terrorism and prevent more of it happening.

That is the primary military objective - to be achieved by targeting facilities belong to the Taleban and to Osama Bin Laden's al-Qaeda network.

The United States would to like to capture or kill Osama Bin Laden himself, but because of the difficulty of achieving that, in public the idea is being played down.

The precondition for everything is to destroy whatever Afghanistan possesses by way of air defences.

Toppling the Taleban?

At the same time, Washington is trying to undermine the ruling Taleban in Afghanistan, both by its own bombing campaign, and by opening the way for the opposition Northern Alliance to make advances on the ground.

That needs careful calibration.

Attack too hard, and it might consolidate support for the Taleban regime.

Give too much support to the Northern Alliance, and that might alienate both Pakistan and opinion elsewhere in Afghanistan.

That explains the cautious formulation by the US Defense Secretary, Donald Rumsfeld, who said they wanted "to alter the military balance over time by denying the Taleban the offensive systems that hamper the progress of the various opposition forces".

It also explains Washington's refusal to be specific about what sort of government it would like to succeed the Taleban.

Nevertheless, officials in Washington are optimistic that the Taleban can be ousted from power.

Winning hearts and minds

President Bush himself needs to persuade domestic opinion in the United States that he is living up to his tough rhetoric.

But he also hopes to keep international opinion on his side, by showing that the US is not targeting civilians in Afghanistan.

At the very opening of his address to the nation, the President spoke of "carefully targeted actions".

The first night of bombing is being combined with airdrops of food.

The ambitious goal is to win the support of the Afghan people, even as their country is under attack.

So it is a delicate balancing act, designed to satisfy opinion in the United States, overseas and even in Afghanistan.

All of that, while Washington continues to insist that its campaign against terrorism is global.

The biggest challenge could simply be living up to the expectations created.