the component of the motion which is perpendicular to the radius vector is considered. In that case, where is the perpendicular component of the motion. Expanding, rearranging, and reducing, angular momentum can also be expressed, where is the length of the moment arm, a line dropped perpendicularly from the origin onto the path of the particle. It is this definition, to which the term moment of momentum refers. Scalar angular momentum from Lagrangian mechanics Another approach is to define angular momentum as the conjugate momentum (also called canonical momentum) of the angular coordinate expressed in the Lagrangian of the mechanical system. Consider a mechanical system with a mass constrained to move in a circle of radius in the absence of any external force field. The kinetic energy of the system is And the potential energy is Then the Lagrangian is The generalized momentum "canonically conjugate to" the coordinate is defined by Orbital angular momentum in three dimensions To completely define orbital angular momentum in three dimensions, it is required to know the rate at which the position vector sweeps out angle, the direction perpendicular to the instantaneous plane of angular displacement, and the mass involved, as well as how this mass is distributed in space. By retaining this vector nature of angular momentum, the general nature of the equations is also retained, and can describe any sort of three-dimensional motion about the center of rotation – circular, linear, or otherwise. In vector notation, the orbital angular momentum of a point particle in motion about the origin can be expressed as: where is the moment of inertia for a point mass, is the orbital angular velocity of the particle about the origin, is the position vector of the particle relative to the origin, and , is the linear velocity of the particle relative to the origin, and is the mass of the particle. This can be expanded, reduced, and by the rules of vector algebra, rearranged: which is the cross product of the position vector and the linear momentum of the particle. By the definition of the cross product, the vector is perpendicular to both and . It is directed perpendicular to the plane of angular displacement, as indicated by the right-hand rule – so that the angular velocity is seen as counter-clockwise from the head of the vector. Conversely, the vector defines the plane in which and lie. By defining