We have to understand the radical Muslims, either to be able to get along with them, or to more efficiently destroy them. A core feature of their belief is the concept of jihad, or holy war.
I approach topics like this systematically. Let me explain my thinking.
1) Not all Muslims want to kill us. Just as not all Texans are cowboys, and not all lettuce is iceberg, we can be wrong in our perceptions. I will say this, however: Those Muslims who do NOT want to kill us are very quiet people.
2) There is a Christian parallel to the view the radical Muslims have. It's the biblical story of Sodom and Gomorrah. According to the bible, the two cities were so wicked that God didn't want them to exist, so he destroyed them. Radical Muslims consider western society so wicked that it must be wiped out.
3) I can see their point. Music, movies, dancing, television -- they are all very coarse now. I'm a crude person from a crude background, yet I'm occasionally shocked at what I see or hear. People with more delicate sensibilities must be really suffering. Additionally, for the last sixty years the United States has gone about its business, poking its nose in some places it didn't belong, neglecting other places where we could have done some good, basically doing what nations do.
4) Having said that, I'm not willing to be killed for the sake of understanding; I don't want my children to die; I don't want my way of life wiped from the face of the planet. And that seems to be the purpose of the radical Muslims.
5) With completely opposite goals -- radical Muslims want Americans dead, and Americans don't want to die -- there is no middle ground, no chance for compromise. Any sort of rapprochement the radical Muslims advance before they are defeated into submission is a distraction. They can not afford to compromise on this fundamental belief that has become so central to their lives.
6) The only original theory I've ever had is this ("original" in that I don't remember hearing it anywhere else): A conflict will be conducted at the lowest level to which either side is willing to go. If I'm arguing with my neighbor, and he begins to punch me in the stomach, I will have to react physically in some way or I won't survive. (Yes, I saw "Gandhi." There are exceptions to the rule.)
7) We need to win in several ways. First, we must inflict such damage and pain to their side that the cost to them is too great to continue. Second, we must convince them that all they have to do to stop the pain is to walk away, go back to their lives, and not kill people. Third, we must influence their society in such a way that they can live in harmony with other societies and religions. (Muslims are killing Buddhists in southeast Asia.)
8) Conversely, anything less than complete American victory, such as leaving Iraq and the rest of the Middle East prematurely, will be perceived as weakness by radical Muslims. This weakness will embolden them, encouraging them to be even more aggressive. This is exactly what Osama bin Laden claimed in an interview, referring to the American withdrawal from Somalia.
The military might of the United States -- Schwarzkopf's "Thunder and Lightning" -- must be brought to bear on the radical Muslims. When a terrorist is in his home, he should be jumpy, listening for the sound of the drone bomb that is seeking him out. His fear of America must override his desire to destroy it. Only then can we move the level of the conflict upward, and eventually eliminate jihad altogether.