Angular momentum

In physics, angular momentum intuitively measures how much the linear momentum is directed around a certain point called the origin; the moment of momentum. Since angular momentum depends upon the origin of choice, one must be careful when discussing angular momentum to specify the origin and not to combine angular momenta about different origins.

The traditional mathematical definition of the angular momentum of a particle about some origin is:

L = r×p

where L is the angular momentum of the particle, r is the position of the particle expressed as a displacement vector from the origin, and p is the linear momentum of the particle. If a system consists of several particles, the total angular momentum about an origin can be gotten by adding (or integrating) all the angular momenta of the constituent particles.

For many applications where one is only concerned about rotation around one axis, it is sufficient to discard the vector nature of angular momentum, and treat it like a scalar where it is positive when it corresponds to a counter clock-wise rotations, and negative clock-wise. To do this, just take the definition of the cross product and discard the unit vector, so that angular momentum becomes:

L = |r||p|sinθ

where θ is the angle between r and p measured from r to p; an important distinction because without it, the sign of the cross product would be meaningless. From the above, it is possible to reformulate the definition to either of the following:

L = ±|p||rperpendicular|

where rperpendicular is called the lever arm distance to p. The easiest way to conceptualize this is to consider the lever arm distance to be the distance from the origin to the line that p travels along. With this definition, it is necessary to consider the direction of p (pointed clock-wise or counter clock-wise) to figure out the sign of L). Equivalently:

L = ±|r||pperpendicular|

where pperpendicular is the component of p that is perpendicular to r. As above, the sign is decided base on the sense of rotation.

In analogy to Newton's second law for linear momentum, we have the following law about angular momentum:

where τ is the net torque about the origin. This implies that angular momentum is a conserved quantity as long as there is no net torque applied to the particle. What's more, this conservation can be generalized to a system of particles under most conditions so that:

where τexternal is any torque applied to the system of particles.

The conservation of angular momentum is used extensively in analyzing what is called central force motion. In central force motion, two bodies form an isolated system not influenced by outside forces, and the origin is placed somewhere on the line between the two bodies. Since any force the bodies exert on each other must be directed along this line, there can be no net torque, with respect to the afore-mentioned origin, on either body. Thus, angular momentum is conserved. Constant angular momentum is extremely useful when dealing with the orbits of planets and satellites, and also when analyzing the Bohr model of the atom.

In modern (late 20th century) theoretical physics, angular momentum is described using an different formalism. Under this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance (As a result, angular momentum isn't defined for general curved spacetimes, unless it happens to be asymptotically rotationally invariant). For a system of point particles without any intrinsic angular momentum, it turns out to be

(Here, the wedge product is used.).




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