Stokes' second problem describes the oscillatory shear flow generated by a tangentially oscillating flat plate in a semi-infinite viscous fluid flow. The velocity satisfies $u_t=\nu u_{xx}$, with $u(0,t)=U_0\cos \omega t$ and decay away from the plate. For $\omega >0$, the displayed time-periodic solution follows by taking the real part of $U_0{e}^{i\omega t-(1+i)x/\delta }$. The length $\delta$ is the penetration depth: both amplitude attenuation and phase lag grow with $x/\delta$. Boundary data alone also allow homogeneous transients; the periodic solution describes the established oscillation after those transients decay.

Generalized forces are the coefficients of virtual work in a set of generalized coordinates. A translational coordinate has a force as its coefficient, while an angular coordinate has a torque. A generalized force conjugate to $q_i$ has units of energy divided by the units of $q_i$. Forces not already included in the Lagrangian enter the Euler-Lagrange equations as $d(\partial L/\partial {\dot{q}}_i)/dt-\partial L/\partial q_i=Q_i$.

An internal photon contribution in a perturbative interaction amplitude; it need not obey the on-shell relativistic energy-momentum relation of a detected free photon.

The Boyle temperature is a temperature at which the second virial coefficient vanishes. At that temperature the compressibility factor has no term linear in number density, but higher terms in the virial expansion need not vanish. For a bounded strictly repulsive finite-range pair potential, $B_2(T)>0$ at every finite positive temperature, with $B_2(T)\to 0$ as $T\to \infty$; calling its Boyle temperature infinite describes this limiting cancellation, not a finite root.

For $H=p^2/(2m)+k{x}^4$ with $k>0$, the normalized even trial wavefunction $f_{\alpha }=(\alpha /\pi {)}^{1/4}{e}^{-\alpha x^2/2}$ has Rayleigh quotientThe Gaussian moment recurrence gives the potential term, and integration by parts gives the kinetic term. Its minimum is at ${\alpha }^3=6mk/{\hslash }^2$, proving the displayed upper bound by the Rayleigh-Ritz variational principle.

An odd trial family $g_{\beta }=(4{\beta }^3/\pi {)}^{1/4}x{e}^{-\beta x^2/2}$ is orthogonal to the exact even ground state. The Sturm oscillation theorem identifies the lowest odd state as the first excited state, so minimizing within this family bounds $E_1$:Orthogonality to an approximate ground trial alone would not justify this excited-state bound; odd parity ensures orthogonality to the exact ground state. Both bounds exhibit the characteristic quartic-oscillator energy scale $({\hslash }^{4}k/m^2{)}^{1/3}$.

For a sphere with constant mass density $\rho$ collecting mass at rate $\beta$ per unit surface area, $\dot{m}=4\pi r^2\beta$ and $m=4\pi \rho r^3/3$ give $\dot{r}=\beta /\rho$. If it falls through stationary material under constant gravitational acceleration $g$ without drag, its velocity satisfies $\dot{v}+3\dot{r}v/r=g$. With $v(0)=0$, an integrating factor $r^3$ gives $v=g[r-r_0^4/r^3]/(4\dot{r})$.

Let fuel of remaining mass $M_f$ be carried at radius $a$ on a rotor whose dry moment of inertia is $I_d$. Then $I=I_d+a^2{M}_f$. Fuel leaves tangentially backwards with speed $v_e$ relative to the rim, so its inertial tangential velocity is $a\omega -v_e$. Conserving angular momentum of rotor plus newly emitted fuel during $d{M}_f<0$ givesSince $dI=a^{2}d{M}_f$, integration yields ${\omega }_f-{\omega }_i=(v_e/a)\log (I_i/I_f)$. Including the fuel's co-rotation is essential: the rotor alone is an open system and its angular momentum is not conserved. For three radial rods of mass $m$ and length $a$, each carrying a point rocket of dry mass $m$ and fuel mass $m_f$, $I_i=(4m+3m_f)a^2$ and $I_f=4m{a}^2$.