origin, is the velocity of the center of mass w.r.t. the origin, is the position vector of particle w.r.t. the center of mass, is the velocity of particle w.r.t. the center of mass, The total mass of the particles is simply their sum, The position vector of the center of mass is defined by, By inspection, and The total angular momentum of the collection of particles is the sum of the angular momentum of each particle, Expanding , Expanding , It can be shown that (see sidebar), and therefore the second and third terms vanish, The first term can be rearranged, and total angular momentum for the collection of particles is finally, The first term is the angular momentum of the center of mass relative to the origin. Similar to , below, it is the angular momentum of one particle of mass M at the center of mass moving with velocity V. The second term is the angular momentum of the particles moving relative to the center of mass, similar to , below. The result is general—the motion of the particles is not restricted to rotation or revolution about the origin or center of mass. The particles need not be individual masses, but can be elements of a continuous distribution, such as a solid body. Rearranging equation () by vector identities, multiplying both terms by "one", and grouping appropriately, gives the total angular momentum of the system of particles in terms of moment of inertia and angular velocity , Single particle case In the case of a single particle moving about the arbitrary origin, and equations () and () for total angular momentum reduce to, Case of a fixed center of mass For the case of the center of mass fixed in space with respect to the origin, and equations () and () for total angular momentum reduce to, Angular momentum in general relativity In modern (20th century) theoretical physics, angular momentum (not including any intrinsic angular momentum – see below) is described using a different formalism, instead of a classical pseudovector. In this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance. As a result, angular momentum is not conserved for general curved spacetimes, unless it happens to be asymptotically rotationally invariant. In classical mechanics, the angular momentum of a particle can be reinterpreted as a plane element: in which the exterior product (∧) replaces the