Sarkozy was not only concerned with his low poll numbers, but he also saw a way to get back at Gadhafi, whom he had tried to "rehabilitate" in late 2007, by hosting him for a week but was rebuffed on oil contract he wanted for France. That would have been a coup for him. Sarkozy was able to convince a skeptical Hillary Clinton of the need "to protect Libyan civilians" and she was able to sell the idea to Obama on the same grounds, especially given his Cairo speech.The Brits were brought on board (this should be no surprise as Cameron was looking for a cause to bolster his support at home, where the economy and his austerity budget was creating panic eg drop in customer confidence of 20%, and of course the Brits are always looking for an excuse to re-assert themselves in the Middle East). Fom the Christian Crusades to the dismantling of the Ottoman Empire, the West led by Britain and France have been trying control the Middle Esat, because they, the Arabs, Persians/Iran, Iraqis, Afghans are incapable of managing their own affairs and that only the Western powers can do so. This Western arrogance /racism has been justified on various grounds...protecting religious sites, the threat of godless communism, the brutality of dictators, the need to provide humanitarian aid, international law/spreading of freedom and human rights. They have used these and other reasons, for example, to arm and insulate the Saudis, to create the state of Iraq (1920's) , as a counter weight to Iran and to carve Kuwait out of it, Lebanon out of Syria and to plant their Trojan Horse, Israel within their midst.
These machinations have not worked. If anything, they have created wars, instability and misery. This new battle for control will also fail. You cannot win control from 30,000 ft.with bombs and even 'boots on the ground"will not work.
The people of the Middle East are shaping their own history, and we, the West should get out of their way.