For a rigidly rotating prescribed vacuum path, use spatial coordinates attached to the apparatus and laboratory time $t$. At a path point $\mathbf{r}$, the local apparatus velocity is $\mathbf{v}=\mathbf{\Omega }\times \mathbf{r}$. A light displacement has instantaneous laboratory displacement $d\mathbf{r}+\mathbf{v}dt$, up to the common rotation of the spatial axes. The null condition is thereforeWrite $d\ell =|d\mathbf{r}|$. To first order in $|\mathbf{v}|/c$, the positive root isReversing the path reverses the dot product term but leaves the leading path-length term unchanged. Subtract the two return times and use the scalar triple product identity:This is a line integral version of the Sagnac area law; the vector area identity also follows from Stokes theorem. For a mirror-defined polygon the actual ray paths move slightly as the mirrors rotate; the stationary optical path at zero rotation gives the same first-order result. The prescribed-path calculation is an idealization, not an exact finite-speed formula for every moving-mirror layout.

The dot product explains the geometry: only the projection of the oriented vector area along the angular velocity contributes. A planar loop whose normal vector is perpendicular to $\mathbf{\Omega }$ has zero leading delay. For repeated traversals with the same orientation, the signed vector areas add; oppositely oriented lobes can cancel even when the total unsigned area is large. Equal perimeter alone does not imply equal response. A constant translational velocity contributes nothing to this leading closed-path line integral, since ${\oint }_{C}d\mathbf{r}=0$.

A Sagnac interferometer uses a beam splitter to send light with optical coherence around a common closed optical path in opposite directions, then recombines the beams to read their phase difference. Mirrors or a guiding path define the loop; the source, beam splitter, loop and readout rotate together in the basic experiment. A small relative beam tilt makes spatial interference fringes whose displacement reveals a change of phase difference. At one fixed output, a photodetector can instead record the corresponding change in light intensity.

At fixed incident flux, projectile energy and detector geometry, the Rutherford differential cross-section gives count rate $S\propto \mathcal{N}{Z}^2$, where $\mathcal{N}$ is the target nuclei per unit area. Equal $\mathcal{N}$ gives a $Z^2$ rate ratio. Equal foil mass per area instead gives $Z^2/A$, with $A$ the atomic mass number, while equal physical thickness also requires the material mass density. An empirical log-log plot with slope $s$ supplies a different, experimentally normalized relation $S\propto A^s$. These normalizations must be specified before comparing different target elements.

The ideal rubber elasticity relation infers active strand number density from the shear modulus. If a quoted modulus is instead Young's modulus, nearly incompressible behavior gives ${\nu }_s\simeq E/(3k_{B}T)$. This is not automatically the junction density: for junction functionality $f$ and no dangling ends or loops, counting strand ends gives ${\nu }_s=f{\nu }_j/2$. Network fluctuations and inactive strands modify the modulus, so converting it into a chemical cross-link count requires a network model.

For $H=-\gamma B_z{S}_z-\gamma B_x{S}_x\cos (\omega t)$, choose ${\omega }_z=\gamma B_z>0$ and nominal resonance $\omega ={\omega }_z$. Starting from up spin, the rotating-wave approximation gives the normalized Bloch vector in the title, with signed $\Omega =\gamma B_x/2$. It follows by transforming the spinor $(\cos (\Omega t/2),i\sin (\Omega t/2){)}^T$ from the interaction picture back to the Schrödinger picture. Thus weak transverse driving combines slow meridional Rabi oscillations with fast Larmor precession.

This result requires $|B_x|\ll |B_z|$ and cannot be extrapolated as a controlled strong-drive formula. The exact equation is $\dot{\mathbf{s}}=\gamma \mathbf{s}\times (B_x\cos \omega t,0,B_z)$. For $|\gamma B_z|t\ll 1$, neglecting the bias gives the controlled leading solutionThe fixed-axis driving generators commute at all times, making this solution exact when the bias is zero at any prescribed drive frequency. At nominal resonance in a strong-drive experiment, the initial rotation scale is $\Theta \simeq \gamma B_{x}t$, twice the incorrectly extrapolated rotating-wave rate. Bias effects must be retained on longer times.

A mechanism in which rotating weights move outward as angular speed increases, allowing their displacement to indicate or regulate a machine's speed. The steady configuration follows the balance of actual applied forces and centripetal acceleration, or equivalently of applied forces and centrifugal force in a rotating reference frame. Geometry and the restoring mechanism determine the speed-displacement relation.

Set the inertial cylindrical angle to $\Phi =\varphi +\omega t$. The Minkowski metric becomes $d{s}^2=(c^2-{\omega }^2{r}^2)d{t}^2-2\omega r^{2}dtd\varphi -d{r}^2-r^{2}d{\varphi }^2-d{z}^2$. For a massive geodesic parametrized by proper time, its cyclic coordinates give conserved covariant four-momentum components $p_t=m[(c^2-{\omega }^2{r}^2)\dot{t}-\omega r^2\dot{\varphi }]$, $p_{\varphi }=-m{r}^2(\dot{\varphi }+\omega \dot{t})$ and $p_z=-m\dot{z}$. Their combination gives $\dot{t}=(p_t-\omega p_{\varphi })/(m{c}^2)$. The radial Euler-Lagrange equation gives the displayed coordinate acceleration, although the free worldline is straight in inertial coordinates. The conserved inertial energy is $p_t-\omega p_{\varphi }$, and the inertial axial angular momentum is $-p_{\varphi }$.

The Robertson-Schrödinger uncertainty relation keeps the real part discarded in the proof of the Robertson uncertainty principle. Write $A^{\prime }=A-⟨A⟩I$ and $B^{\prime }=B-⟨B⟩I$. Their quantum covariance is $C_{AB}=\mathrm{Re}⟨A^{\prime }\psi |B^{\prime }\psi ⟩$. On the same finite-moment and product operator domains used above, the Cauchy-Schwarz inequality givesFor position and momentum, form the real, symmetrized covariance matrixA larger coordinatewise product can therefore encode a tilted covariance ellipse, rather than a larger determinant. This explains why the equality case of the Heisenberg uncertainty relation is more restrictive than equality in the Robertson-Schrödinger uncertainty relation.