The examples and perspective in this article deal primarily with the United States and do not represent a worldwide view of the subject. (May 2013)
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The definition of a CoG is "the source of power that provides moral or physical strength, freedom of action, or will to act." Thus, the center of gravity is usually seen as the "source of strength".
The United States Army tends to look for a single center of gravity, normally in the principal capability that stands in the way of the accomplishment of its own mission. In short, the army considers a "friendly" CoG as that element--a characteristic, capability, or locality--that enables one's own or allied forces to accomplish their objectives. Conversely, an opponent's CoG is that element that prevents friendly forces from accomplishing their objectives.
For example, according to US Army Counterinsurgency Field Manual 3-24, the center of gravity in a counterinsurgency is the protection of the population that hosts it.