Angular momentum is a physical quantity describing motion about a chosen point or axis. In classical mechanics, a particle’s angular momentum is the cross product of its position relative to that point and its linear momentum. Extended bodies have angular momentum associated with both their overall motion and their rotation. In quantum mechanics, angular momentum also includes intrinsic spin. Its conservation connects rotational dynamics with the symmetry of physical laws under rotations. (openstax.org)
Classical definition
For a particle with position vector (\mathbf r), measured from a specified origin, and momentum (\mathbf p), angular momentum is
[ \mathbf L=\mathbf r\times\mathbf p. ]
The cross product makes (\mathbf L) perpendicular to the plane containing (\mathbf r) and (\mathbf p), with direction determined by the right-hand rule. Its magnitude is (L=rp\sin\theta), where (\theta) is the angle between those vectors. Consequently, a particle moving along a straight line can have nonzero angular momentum about a point away from that line; circular motion is not required. (openstax.org)
For a nonrelativistic particle of mass (m), (\mathbf p=m\mathbf v). In the International System of Units, angular momentum has units (\mathrm{kg,m^2,s^{-1}}). Its value generally depends on the chosen origin: shifting the origin by (\mathbf a) gives (\mathbf L'=\mathbf L-\mathbf a\times\mathbf p). For a system of particles, the corresponding change involves the system’s total momentum. (openstax.org)
Angular momentum is an axial vector, rather than an ordinary polar vector. Under spatial inversion, position and momentum both reverse, while their cross product remains unchanged. This distinction matters when describing reflection symmetry and parity. (ocw.mit.edu)
Systems and rigid bodies
The total angular momentum of a particle system is
[ \mathbf L=\sum_i\mathbf r_i\times\mathbf p_i. ]
It separates into an orbital contribution from the motion of the center of mass and an internal contribution measured relative to that center:
[ \mathbf L=\mathbf R_{\rm cm}\times\mathbf P+\mathbf L_{\rm cm}. ]
Here (\mathbf P) is total linear momentum. This decomposition distinguishes, for example, a body’s movement around an external point from its rotation about its own center. (mitp-content-server.mit.edu)
For a rigid body, rotational angular momentum about its center of mass satisfies
[ \mathbf L_{\rm cm}=\mathbf I\boldsymbol\omega, ]
where (\boldsymbol\omega) is angular velocity and (\mathbf I) is the inertia tensor. Angular momentum and angular velocity need not be parallel. Along a principal axis, the relation reduces to (L=I\omega), with (I) the moment of inertia about that axis. The distribution of mass, not merely its total amount, therefore determines the angular momentum associated with a given rotation. (mitp-content-server.mit.edu)
Torque and conservation
About a fixed origin in an inertial reference frame, the rate of change of angular momentum equals the net torque:
[ \frac{d\mathbf L}{dt}=\boldsymbol\tau_{\rm ext}. ]
For a particle, torque is (\mathbf r\times\mathbf F), where (\mathbf F) is the applied force. For an ordinary mechanical system whose internal torques cancel, only external torque changes its total angular momentum. If the net external torque is zero, the total angular-momentum vector remains constant, although individual parts may exchange angular momentum. (openstax.org)
A spinning skater illustrates this principle approximately. Bringing the arms inward reduces moment of inertia and increases angular velocity, so that (I\omega) remains nearly unchanged. Conservation of angular momentum does not imply conservation of rotational kinetic energy: the skater’s muscles perform work, changing the energy while external torque remains small. (openstax.org)
A torque can also change angular momentum’s direction. In a rapidly spinning gyroscope, gravitational torque produces precession, in which the spin axis moves around another axis. The familiar slow-precession formula assumes that angular momentum is dominated by the rapid spin; it is not an unrestricted description of every spinning top. (openstax.org)
Rotational symmetry
Noether’s theorem supplies a deeper explanation of conservation. If a system’s action is unchanged by continuous spatial rotations, the corresponding angular momentum is conserved. Rotational invariance means that the dynamics do not depend on an absolute orientation in space; it does not require the moving object itself to have a spherical shape. Symmetry under rotations about just one axis gives conservation of the corresponding angular-momentum component. (mitp-content-server.mit.edu)
For a particle subject to a central force, the force points along the radius from the force center, so its torque about that center vanishes. The conserved angular momentum confines a nonradial trajectory to a plane and implies a constant rate of sweeping out area, linking angular-momentum conservation to orbital mechanics. (ocw.mit.edu)
Quantum angular momentum
In quantum theory, orbital angular momentum is represented by the operator (\hat{\mathbf L}=\hat{\mathbf r}\times\hat{\mathbf p}). Spin angular momentum is intrinsic and is not described as the literal rotation of a classical particle. Both obey the same angular-momentum algebra. For a general angular momentum (\hat{\mathbf J}),
[ [\hat J_x,\hat J_y]=i\hbar\hat J_z, ]
with cyclic counterparts. Here (\hbar=h/(2\pi)), where (h) is the Planck constant. The noncommuting components cannot generally possess simultaneous definite values, whereas (\hat J^2) and one component can. (ocw.mit.edu)
Their simultaneous eigenstates satisfy
[ \hat J^2|j,m\rangle=\hbar^2j(j+1)|j,m\rangle, \qquad \hat J_z|j,m\rangle=\hbar m|j,m\rangle. ]
The quantum number (j) is a nonnegative integer or half-integer, and (m=-j,-j+1,\ldots,j). Orbital angular momentum has integer (j), conventionally written (\ell); intrinsic spin can be half-integer. Thus the magnitude associated with definite (j) is (\hbar\sqrt{j(j+1)}), not simply (j\hbar). (ocw.mit.edu)
Angular momenta combine by operator addition. For two contributions with quantum numbers (j_1) and (j_2), the allowed total values run from (|j_1-j_2|) to (j_1+j_2) in unit steps. This framework describes the coupling of orbital motion and spin in atomic states. (ocw.mit.edu)