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The classical definition of angular momentum is =.The quantum-mechanical counterparts of these objects share the same relationship: = where r is the quantum position operator, p is the quantum momentum operator, × is cross product, and L is the orbital angular momentum operator.
The trivial case of the angular momentum of a body in an orbit is given by = where is the mass of the orbiting object, is the orbit's frequency and is the orbit's radius.. The angular momentum of a uniform rigid sphere rotating around its axis, instead, is given by = where is the sphere's mass, is the frequency of rotation and is the sphere's radius.
The orbital angular momentum of light (OAM) is the component of angular momentum of a light beam that is dependent on the field spatial distribution, and not on the polarization. OAM can be split into two types. The internal OAM is an origin-independent angular momentum of a light beam that can be associated with a helical or twisted wavefront.
For reference and background, two closely related forms of angular momentum are given. In classical mechanics, the orbital angular momentum of a particle with instantaneous three-dimensional position vector x = (x, y, z) and momentum vector p = (p x, p y, p z), is defined as the axial vector = which has three components, that are systematically given by cyclic permutations of Cartesian ...
The total angular momentum of light consists of two components, both of which act in a different way on a massive colloidal particle inserted into the beam. The spin component causes the particle to spin around its axis, while the other component, known as orbital angular momentum (OAM), causes the particle to rotate around the axis of the beam.
Proof. For an elliptic orbit with specific angular momentum h given by = = we use the general form of the specific orbital energy equation, = with the relation that the relative velocity at periapsis is = = = () = () Thus our specific orbital energy equation becomes = [()] = [() (+) ()] = [+ ()] = [()] and finally with the last simplification we obtain: =
In celestial mechanics, the specific relative angular momentum (often denoted or ) of a body is the angular momentum of that body divided by its mass. [1] In the case of two orbiting bodies it is the vector product of their relative position and relative linear momentum , divided by the mass of the body in question.
Orbital position vector, orbital velocity vector, other orbital elements. In astrodynamics and celestial dynamics, the orbital state vectors (sometimes state vectors) of an orbit are Cartesian vectors of position and velocity that together with their time () uniquely determine the trajectory of the orbiting body in space.