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For a circular fixed-mass planet, $L=M_p{a}^2\Omega$ and $\dot{a}=2\Gamma /(M_{p}a\Omega )$. With $\Gamma =C(q/h{)}^2\Sigma a^4{\Omega }^2$, the radial migration timescale is $a/|\dot{a}|=(M_{\star }/M_p)[M_{\star }/(\Sigma a^2)]h^2/(2|C|\Omega )$. The factor of two distinguishes radial evolution from $L/|\Gamma |$. The dimensionless coefficient and its sign depend on the disk model.

A neutrino with energy much larger than its mass. At fixed momentum, expanding the relativistic energy-momentum relation gives $E_i\simeq p+m_i^2/(2p)$, so differences of propagation phases are governed by $\Delta m^2/(2E)$.

For one unit-charge Dirac field in $d=4-2ϵ$, define the insertion as $i{\Pi }_{\mu \nu }=i(q^2{g}_{\mu \nu }-q_{\mu }{q}_{\nu })\Pi$. The one-loop ultraviolet pole is ${\Pi }_{\mathrm{UV}}=-e^2/(12{\pi }^2ϵ)$. The Ward identity cancels any apparent divergent vector mass term. With photon counterterm insertion $-i(q^2{g}_{\mu \nu }-q_{\mu }{q}_{\nu }){\delta }_3$, this gives ${\delta }_3=-e^2/(12{\pi }^2ϵ)$ in the minimal subtraction scheme.

For internal momentum $q\gg k$, the P22 contribution to the one-loop matter power spectrum starts at $k^4$, while twice the P13 contribution to the one-loop matter power spectrum starts at $k^{2}P(k)$. The latter has the same momentum dependence as the effective sound-speed counterterm in large-scale structure. Short-scale stochastic contributions start at $k^4$ by mass and momentum conservation.

At fixed external momentum $\mathbf{k}$, mass and momentum conservation implyas $q\to \infty$. In particular, the constant term in the hard-momentum expansion must cancel.

Underdamped Langevin dynamics augments position by momentum and combines Hamiltonian transport with friction and matching Gaussian noise. For potential $U$, momentum variance $\eta$, and friction $\gamma$, its invariant density is proportional to $\exp [-U(x)-|p{|}^2/(2\eta )]$.

For kernel ${\kappa }^{-1}{p}_{\perp }^2+{\mu }^{-1}{p}_{\parallel }^4+m^2$ with positive stiffnesses, preserve both derivative coefficients by $q_{\perp }=b{p}_{\perp }$ and $q_{\parallel }=b^{1/2}{p}_{\parallel }$. The Fourier field multiplier is $\tilde{Z}=b^{(2D+3)/4}$, the mass becomes $bm$, and a uniform source becomes $b^{(2D+3)/4}h$. The anisotropic effective dimension is $D-1/2$ and the Gaussian exponents are $\alpha =9/4-D/2$, $\Delta =(2D+3)/8$.

A uniform circular disc of radius $a$ and mass $m$, pivoted on a perpendicular axis through its rim, has moment of inertia $I_A=3m{a}^2/2$ by the parallel axis theorem. Its hanging small oscillations have period $2\pi \sqrt{3a/(2g)}$. Measured from the upright position, the falling equation is $\ddot{\theta }=2g\sin \theta /(3a)$. Exact upright rest remains an unstable equilibrium; the energy level of that state describes a limiting separatrix after perturbation, with ${\dot{\theta }}^2=4g(1-\cos \theta )/(3a)$. The center acceleration and pivot reaction follow by resolving radial and tangential components.

Inside a homogeneous spherical mass, gravity is proportional to position. Projection onto a fixed straight chord gives $\ddot{y}=-{\omega }_0^{2}y$, independently of its perpendicular offset from the sphere centre. Release from one surface endpoint at rest reaches the opposite endpoint after $\pi /{\omega }_0$. The transverse force is supplied by the tunnel constraint. This idealization neglects removed tunnel mass, rotation, density variation and dissipative forces.

For fixed mass $M$, a uniform sphere of physical radius $R$ has $W_G=-3G{M}^2/(5R)$ and $W_{\Lambda }=-\Lambda M{R}^2/10$, using the convention in which its outward acceleration is $\Lambda r/3$. The virial theorem reads $2T+W_G=2W_{\Lambda }$. Defining $x=R_f/R_{\mathrm{ta}}$ and $\eta =\Lambda /(4\pi G{\rho }_{\mathrm{ta}})$, conservation of energy gives $2\eta x^3-(2+\eta )x+1=0$. The collapsing branch has $x=1/2-\eta /8+O({\eta }^2)$. For small positive cosmological constant, it is more compact at fixed turnaround data, even though the outward force inhibits turnaround and collapse.

For $N$ Dirac fields with common mass, $\mathcal{L}=\bar{\Psi }(i/\partial -m{1}_N)\Psi$ is invariant under constant complex-linear rotations $\Psi \mapsto U\Psi$ with $U\in U(N)$. A Hermitian flavour generator $T^A$ has Noether current $j_A^{\mu }=\bar{\Psi }{\gamma }^{\mu }{T}^A\Psi$. A nondegenerate fermion mass matrix restricts the symmetry to unitary matrices commuting with it. If the theory is entirely free and rotations into charge-conjugate fields are allowed, its enhanced internal symmetry of free Dirac fields can be larger than this subgroup.

In simple hierarchical models, increasing system mass raises the virial temperature and can carry primordial gas beyond its efficient atomic line-cooling interval. At assembly densities, the cooling-time constraint then distinguishes galaxy-sized condensations from larger hot groups or clusters. The resulting characteristic mass is approximate, depends on composition and structure, and does not prohibit subsequent mergers of already formed stellar systems.

The $SU(2)$ flavor symmetry rotating down and strange quarks, with the up quark a singlet. Since down and strange quarks have equal electric charge, the charge operator commutes with all U-spin generators, and every irreducible U-spin multiplet has a common charge. Unequal down- and strange-quark masses break this symmetry.

In a uniform U-tube containing incompressible liquid of density $\rho$, cross section $A$ and total moving length $l$, let one surface fall by $y$ while the other rises by $y$. The height difference is $2y$, and potential energy relative to equilibrium is $U=\rho gA{y}^2$. The whole moving mass is $\rho Al$, so $K=\rho Al{\dot{y}}^2/2$. Newton's second law or energy conservation gives $\ddot{y}+(2g/l)y=0$ and ${\omega }_0=\sqrt{2g/l}$. This assumes small excursions that keep the free surfaces in the uniform vertical limbs.

In $c=\hslash =1$ units, vacuum energy density has mass dimension four, Newton constant dimension minus two, and ${\Lambda }_{\mathrm{geom}}=8\pi G{\rho }_{\mathrm{vac}}$ dimension two. A value quoted in $\mathrm{Ge}{\mathrm{V}}^4$ is an energy density, not the geometric cosmological constant.

The rest mass of a stable soliton is the lowest energy in its topological sector minus the vacuum energy, in the infinite-volume limit. If Euclidean kernels $K_Q(\tau )$ have nonzero overlap with the sector's lowest-energy states, then $M=-{\lim }_{\tau \to \infty }{\tau }^{-1}\log [K_1(\tau )/K_0(\tau )]$, followed by the infinite-volume limit. A Euclidean path integral represents these kernels, with boundary wavefunctionals included if necessary.

When a vacuum vessel's window shatters, a nearly atmospheric initial pressure difference gives inward pressure force $\Delta pA$ on frontal area $A$. Dividing by the glass mass $M$ estimates its initial collective acceleration. An ideal atmospheric pressure-work budget $p_{0}V$ for vessel volume $V$ would give a collective kinetic energy estimate $M{v}_{\mathrm{rms}}^2/2\lesssim p_{0}V$ if assigned entirely to the fragments. This is a model-dependent energy budget, not a universal bound on every fragment's speed: air motion, fragmentation, pressure equalization and wall impacts alter the transfer, and unequal sharing can accelerate a small fragment faster than the collective estimate.