Michael Ramsey at Opinio Juris ("a forum for informed discussion and lively debate about international law and international relations") offers his learned opinion.
Money quote(s):
"Every major figure from the founding era who commented on the matter said that the Constitution gave Congress the exclusive power to commit the nation to hostilities. Notably, this included not only people with reservations about presidential power, such as James Madison and Thomas Jefferson, but also strong advocates of the President’s prerogatives, such as George Washington and Alexander Hamilton. As President, Washington on several occasions said that he could not undertake offensive military actions without Congress’ approval. Hamilton is especially significant, because his views on the need for a strong executive went far beyond those of his contemporaries. Yet Hamilton made it very clear that he read the Constitution not to allow the President to begin a war – as he put it at one point, “it belongs to Congress only, to go to war.” "
This seems clear enough. Congress has subsequently, during the Vietnam "Conflict" undertaken to delegate a little bit of that power across branches of government to the presidency, in the War Powers Act. But only within certain expressly circumscribed parameters.
"The fact that our use of force is limited to air strikes should not matter. Limited wars were well-known in the 18th century (Britain and France fought a limited war at sea and in North America during the American Revolution). The U.S. fought two limited wars early in its history, against France beginning in 1798 and against Tripoli in 1801. So far as I know, every person commenting on these events at the time thought that Congress had to authorize any initiation of force, even limited naval attacks. (In 1801, Hamilton argued that authorization wasn’t needed because Tripoli, not the U.S., began the war; but he agreed that congressional authorization would otherwise be necessary even in the context of an attack on a single Tripoli warship.)
Thus the founding generation thought the Constitution reserved war-initiation power to Congress. How could this be, though, if Congress has only the power to “declare War”, which we may think refers to making a (now-outmoded) formal announcement? Why can’t the President begin a war informally, merely by ordering an attack, without a declaration?
The answer is that in founding-era terminology war could be “declared” either by formal announcement or by military action initiating hostilities."
Fair enough. Deeds then, when words would be superfluous.
"Perhaps, though, the President also has power to declare war (after all, the Constitution expressly says only that Congress has it, not that the President doesn’t, and it could be part of the President’s power as commander-in-chief). Returning to Hamilton, a key passage in his Federalist 32 argued that often constitutional power could be held concurrently by different entities. But, he continued, an exclusive grant of power would arise where concurrent power would be “totally contradictory and repugnant” – that is, when one branch’s exercise of a power would wholly undermine an express grant to another branch. Hamilton didn’t give the example of declaring war here, but it fits his model: war, once launched, cannot be undone without consequences. If Congress’ power is to decide when war should begin, it follows that the President cannot independently launch attacks."
That's one of the (several) drawbacks of the War Powers Act; aside from the "nuclear options" of denying funding to American troops under fire and engaged in mortal combat OR initiating impeachment proceedings against, variously, the president and/or the secretaries of defense and state.
"(T)he founding generation’s views are clear and have firm basis in the Constitution’s text: the declare war clause gives Congress the exclusive power to decide when war should be “declared” – meaning begun by “word or action.” In Libya, President Obama has “declared” a war – a limited one, to be sure, but still a war by 18th century definitions – without congressional approval. That contravenes both the Constitution’s text and the founding era’s consensus understanding."
So what is to be done? It would seem that "standing" to do anything rests with two bodies: the Congress and the electorate. It remains to be seen which one, if either, will act, if at all. Neither Congress nor the public wish to deny support to our soldiers, airmen, and sailors who are doing their jobs and going into harm's way. Been there, done that. But note the lack of any "bump" in the president's poll standings.
People (and congressmen) are waiting, giving the benefit of the doubt, but when or if the U.S. begins to take casualties, particularly if "mission creep" occurs and there are "boots on the ground," then all bets are off.