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The principal moments of inertia about a specified point $O$ are the eigenvalues $I_1,I_2,I_3$ of the inertia tensor ${\mathbf{I}}_O$. Their associated unit vectors that are eigenvectors specify principal axes through $O$. The spectral theorem for real symmetric matrices guarantees an orthonormal basis of these eigenvectors, even for an irregular rigid body. Choose their signs to make a right-handed basis and collect them as the columns of an orthogonal matrix $Q$:Thus diagonalization removes the off-diagonal coupling in the rotational kinetic energy. The eigenvalues belong to the mass distribution and reference point, independently of which orthonormal basis initially describes them. ^[ref]

In this principal body frame, write angular velocity as $\mathbf{\omega }={\sum }_i{\omega }_i{\mathbf{e}}_i$. ThenUse these as total angular momentum and kinetic energy only under the reference-point conditions stated under inertia tensor. Along a single principal axis with $I_i>0$, angular momentum is parallel to angular velocity. For a mixture of directions with different principal moments of inertia, it generally is not. A principal axis need not be a geometric symmetry axis: it is selected by the complete mass distribution. A principal axis also need not give stable free rotation; for three distinct principal moments of inertia, the intermediate axis theorem supplies the stability distinction.

For any unit vector $\mathbf{n}={\sum }_i{n}_i{\mathbf{e}}_i$,The axial moment of inertia is a weighted average of the principal moments of inertia. At fixed angular speed, the least and greatest rotational kinetic energies occur in the corresponding eigenspaces. At fixed nonzero angular momentum magnitude $L$, instead, $K=\frac{1}{2}{\sum }_i{L}_i^2/I_i$, so the largest principal moment of inertia gives the smallest rotational kinetic energy. These comparisons hold the reference point fixed.

There is an additional physical restriction beyond positive semidefiniteness. In principal axes coordinates,with two analogous cyclic inequalities. Therefore three arbitrary positive numbers do not always represent a possible inertia tensor. For a distribution in the $xy$ plane through $O$, $I_3=I_1+I_2$, as in the planar inertia tensor of paired point masses. A uniform solid ellipsoid of mass $M$ and semiaxes $a,b,c$ supplies a continuous three-dimensional example: using its mass second-moment tensor, its central principal moments of inertia are $M(b^2+c^2)/5$, $M(a^2+c^2)/5$, $M(a^2+b^2)/5$. Setting $a=b=c=R$ recovers the moment of inertia of a uniform solid sphere.

A rapidly spinning magnetosphere adds angular momentum to inflowing matter outside the corotation radius, inhibiting accretion and potentially ejecting gas. The centrifugal onset condition does not by itself ensure that every parcel escapes to infinity.

A missing proton relative to a filled nuclear shell. In a closed-core approximation a single hole in a $j$ orbital has the same total angular momentum $j$ and orbital parity as the removed particle. For example, removing a $p_{1/2}$ proton from a spin-zero closed core gives $J^P=1/2^{-}$.

An axial vector transforms as ${\mathbf{a}}^{\prime }=(\det Q)Q\mathbf{a}$ under an orthogonal matrix $Q$. It agrees with a polar vector under a proper rotation matrix and has the additional determinant sign under an orientation-reversing transformation. Angular momentum and angular velocity are axial vectors.

Loss of a rotating neutron star's angular momentum and rotational energy through electromagnetic torque and a magnetized outflow. For constant moment of inertia $I$, differentiating $E_{\mathrm{rot}}=I(2\pi /P{)}^2/2$ gives the displayed positive loss rate when $\dot{P}>0$. It is not the same as the measured radiation luminosity in any one observing band.

The rovibrational spectroscopy branch with unchanged rotational angular momentum quantum number. Differences between upper and lower rotational constants can resolve its transitions into closely spaced lines. Allowed lower-state values depend on the angular momenta of both states.

Put $B_j=[m(x)(x-\bar{x}{)}^j{]}_0^L$, $M_j=\int m(x)(x-\bar{x}{)}^{j}dx$, $A=\int \alpha (t)dt$, and $C=B_0(B_2-M_1)-B_1(B_1-M_0)$. For quadratic bending $h_0=-\alpha (t)(x-\bar{x}{)}^2$, the linear recoil correction in elongated-body theory with a stable post-stroke relaxation gives the displayed angle. It follows by setting time derivatives to zero in the integrated lateral and angular momentum equations. The simpler $2UA$ is exact in the zero-body-mass limit, or when $B_1=M_0$, but is not a generic finite-body-mass identity. For $L=1$, $\bar{x}=1/2$, and $m(x)=m_{\ast }{x}^2$, it becomes $2UA/(1+2M_b/m_{\ast })$. Body deformation inertia affects derivative forcing during the stroke, not this final integral relation when the deformation and its time derivative return to zero.

A quantum time-reversal operator is an antiunitary operator reversing spatial momentum and angular momentum. For a time-reversal-invariant Hamiltonian operator, its conjugation of $i$ makes it intertwine forward and backward unitary time evolution without reversing the energy spectrum. A scalar momentum eigenstate may transform as $\hat{T}|\mathbf{p}⟩=e^{i\chi (\mathbf{p})}|-\mathbf{p}⟩$. With unit-modulus phases and a complete normalized basis, expansion of arbitrary states proves antiunitarity.

A quantum vortex has integer winding number of a condensate's complex argument. For atomic mass $m$, its superfluid velocity gives circulation $2\pi \hslash q/m$ for integer $q$. A straight vortex has a density-depleted core and, outside it, tangential velocity $q\hslash /(mr)$. Its long-range flow energy is proportional to $q^2\log (R/a)$, while its angular momentum has the sign of $q$.

A quarkonium is a bound state of a heavy quark and its own antiquark. Examples are charmonium and the bottom-antibottom states called Upsilon mesons. Spectroscopic labels $n{}^{2S+1}{L}_J$ distinguish radial excitation, total quark spin, relative orbital angular momentum and total angular momentum.

For two equal point masses $m$ joined by a light tether of length $l$, the central moment of inertia is $m{l}^2/2$. With conserved angular momentum $L$, the rotational kinetic energy is $L^2/(m{l}^2)$. Slow, smooth contraction gives ${\dot{E}}_{\mathrm{rot}}=-T\dot{l}$ with the displayed leading tension. The exact radial correction is $E_{\mathrm{rad}}=m{\dot{l}}^2/4$ and $T=2L^2/(m{l}^3)-m\ddot{l}/2$, so the total work-energy theorem remains exact for non-quasi-static schedules too.

For test particles with linear drag, negligible radial gas flow and negligible dust feedback, Sub-Keplerian rotation removes solid-particle angular momentum. The displayed inward velocity uses Stokes number $\mathrm{St}={\Omega }_K{t}_s$ and the pressure-gradient parameter of a protoplanetary disk. Drift is fastest at $\mathrm{St}=1$; the physical particle size at that point depends on the local gas and drag law.

For central potential $V(r)=\beta r^k$ and nonzero angular momentum $J$, a circular orbit satisfies $\beta k{r}_0^{k+2}=J^2/m$, so attraction requires $\beta k>0$. Twice differentiating the radial effective potential $V+J^2/(2m{r}^2)$ gives $V_{\mathrm{eff}}^{\prime \prime }(r_0)=(k+2)J^2/(m{r}_0^4)$. Its harmonic approximation therefore has radial angular frequency ${\omega }_r=\sqrt{k+2}|J|/(m{r}_0^2)$, whereas $|{\omega }_{\theta }|=|J|/(m{r}_0^2)$. This proves the ratio and shows stability precisely when $k>-2$. At $k=-2$ the tuned effective potential is flat; at $k<-2$ the radial disturbance grows exponentially. Rationality of this local frequency ratio is only a necessary condition for all nearby orbits to close; Bertrand's theorem makes a stronger finite-amplitude assertion.

For $\mathbf{F}=-A\mathbf{r}$, the turning radii obey $L^2=mA{r}_{\min }^2{r}_{\max }^2$. A radial impulse preserves angular momentum. Its circular orbit therefore has $r_c^4=L^2/(mA)$, or $r_c=\sqrt{r_{\min }{r}_{\max }}$. Cancel the radial velocity where the original orbit crosses $r_c$.

A radial impulse that multiplies the initial circular kinetic energy by $\beta >1$ adds radial velocity $\pm \sqrt{GM/r_0}\sqrt{\beta -1}$ and leaves specific angular momentum $j^2=GM{r}_0$. The new specific orbital energy is $GM(\beta -2)/(2r_0)$, yielding orbital eccentricity $e=\sqrt{\beta -1}$ and semilatus rectum $p=r_0$. For $1<\beta <2$, the apsidal radii are $r_0/(1\pm e)$; for $\beta \ge 2$ there is no finite apocentre. A nonzero radial kick occurs away from an apsis, so the old circular radius is not automatically the new periapsis.

A radial null geodesic in a spherically symmetric spacetime has zero angular momentum and fixed angular coordinates. Its projection is a null curve of the two-dimensional radial Lorentzian metric.

Every maximal radial timelike geodesic in positive-mass Kruskal spacetime is incomplete in at least one direction. Its equations are ${\dot{r}}^2=E^2-1+2M/r$ and $\ddot{r}=-M/r^2$. A finite turning point is a maximum, and each trajectory reaches $r=0$ at one or both ends. The remaining proper time is finite because $d\tau \sim \sqrt{r/(2M)}dr$. Complete circular timelike geodesics exist, but they have nonzero angular momentum.

The rovibrational spectroscopy branch in which absorption raises the total rotational angular momentum quantum number by one. A label $R(J)$ conventionally specifies the lower-state value $J$.

For unpolarized entrance particles, average over their $(2J_a+1)(2J_b+1)$ spin orientations and sum over the isolated resonance's $2J_R+1$ magnetic substates. The resulting weight multiplies the partial-wave cross section. It can exceed one because orbital angular momentum participates; it is not a standalone probability of forming a state. Polarization or several unresolved resonance spins requires more detailed channel counting.

If two nearby inner planets couple much more strongly to each other than to a distant third planet, their slow secular eigenmode has a common complex inclination. With $L_{12}=L_1+L_2$, the effective inner-to-outer coupling isAngular momentum balance makes the reverse coupling $c{L}_{12}/L_3$. The reduced matrix has eigenvalues $0$ and $-c(1+L_{12}/L_3)$. If $L_3\gg L_{12}$, the nonzero frequency is approximately $-c$, while the fast inner differential mode has frequency $-(B_{12}+B_{21})$.