page_title: Angular momentum
section_title: Angular momentum (modern definition)
hierarchical_section_title: Angular momentum / Angular momentum (modern definition)
context_page_description: In physics, angular momentum is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a closed system remains constant.
  In three dimensions, the angular momentum for a point particle is a pseudovector r × p, the cross product of the particle's position vector r and its momentum vector; the latter is p = mv in Newtonian mechanics. This definition can be applied to each point in continua like solids or fluids, or physical fields. Unlike momentum, angular momentum does depend on where the origin is chosen, since the particle's position is measured from it.
  Just like for angular velocity, there are two special types of angular momentum: the spin angular momentum and orbital angular momentum. The spin angular momentum of an object is defined as the angular momentum about its centre of mass coordinate. The orbital angular momentum of an object about a chosen origin is defined as the angular momentum of the centre of mass about the origin. The total angular momentum of an object is the sum of the spin and orbital angular momenta.
context_section_description: 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 cross product × (these products have similar characteristics but are nonequivalent). This has the advantage of a clearer geometric interpretation as a plane element, defined from the x and p vectors, and the expression is true in any number of dimensions (two or higher). In Cartesian coordinates:
  or more compactly in index notation:
  The angular velocity can also be defined as an antisymmetric second order tensor, with components ωᵢⱼ. The relation between the two antisymmetric tensors is given by the moment of inertia which must now be a fourth order tensor:
  Again, this equation in L and ω as tensors is true in any number of dimensions. This equation also appears in the geometric algebra formalism, in which L and ω are bivectors, and the moment of inertia is a mapping between them.
  In relativistic mechanics, the relativistic angular momentum of a particle is expressed as an antisymmetric tensor of second order:
  in the language of four-vectors, namely the four position X and the four momentum P, and absorbs the above L together with the motion of the centre of mass of the particle.
  In each of the above cases, for a system of particles, the total angular momentum is just the sum of the individual particle angular momenta, and the centre of mass is for the system.
caption: The 3-angular momentum as a bivector (plane element) and axial vector, of a particle of mass m with instantaneous 3-position x and 3-momentum p.