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For bound old Kepler orbits, $E<0$, $\lambda <0$ and $\bar{E}>0$ give a positive radius, positive isochrone scale $b$, and positive transformed mass. The shifted energy is $\bar{E}-k=\bar{E}/\lambda <0$. For unbound old orbits with $E>0$, $1/2<\lambda \le 1$ gives an attractive unbound isochrone branch. The case $\lambda =0$ is handled by the separate Kepler–harmonic radial duality.

An exact axial Killing vector field makes Komar angular momentum independent of the surrounding vacuum surface. Therefore gravitational waves in a configuration with exact axisymmetry carry no net angular momentum about that symmetry axis, although they can carry energy. This is specific to axial angular momentum; it is not an absence of radiation.

For three vectors $a,b,c$ in an inner-product space, expansion of each squared norm provesTherefore the second-order backward differentiation formula $3U^{n+2}-4U^{n+1}+U^n=2kL{U}^{n+2}$ with a dissipative operator $L$ decreases the discrete energy ${\mathcal{E}}_n=\Vert U^{n+1}{\Vert }^2+\Vert 2U^{n+1}-U^n{\Vert }^2$. This quadratic form is equivalent to the product norm of the two time levels with constants independent of $k$ and $L$. It proves uniform stability even when amplification roots of a polynomial coalesce inside the unit disk, where an eigenvector-separation proof may lose its bound.

A rigid Skyrmion orientation is quantized on a cover of its Skyrmion collective-coordinate orbit. A lifted combined rotation in the static field's stabilizer subgroup identifies the same classical field and imposes ${\hat{D}}^J(R){\hat{D}}^I(A)\Psi ={\chi }_{\mathrm{FR}}\Psi$, where the sign is the chosen Finkelstein-Rubinstein constraints character of that configuration-space loop. Choosing the nontrivial character models unit baryons as fermions. A $2\pi$ spatial rotation or isorotation of charge $B$ then has sign $(-1{)}^B$: deforming the field to separated unit lumps adds the unit-lump loop classes in the fundamental group ${\mathbb{Z}}_2$, while the labelled orbital paths can be contracted in three dimensions. The resulting Finkelstein-Rubinstein constraints sign is the product of the unit-lump signs. Hence spin and isospin are half-integer for odd $B$ and integer for even $B$. Further stabilizer constraints restrict which pairs $(J,I)$ and which body-fixed states occur. The energy operator comes from the inertia tensors on the orbit; symmetry selects allowed states but does not by itself determine their energies. A bosonic choice of the trivial topological character is mathematically possible but does not model a fermionic nucleon.

Convection transports properties such as energy or composition through bulk fluid motion. Thermal convection is driven by temperature-dependent buoyancy.

For $y^{\prime \prime }+y=\epsilon (y^{\prime }{)}^3$ with positive $\epsilon$, the oscillator energy increases at rate $\epsilon (y^{\prime }{)}^4$. The amplitude-phase equations give $R(T)=R_0(1-3R_0^{2}T/4{)}^{-1/2}$. The weakly nonlinear expansion fails when the growing amplitude makes $\epsilon R^2$ large, so its formal finite-time divergence does not itself control the exact late-time trajectory.

Delayed cooling preserves a young giant planet's large radius by reducing the escape of internal energy. Enhanced atmospheric opacity, an irradiation-maintained outer radiative blanket, and composition gradients that inhibit convection can slow Kelvin-Helmholtz contraction. These mechanisms retain existing heat rather than supplying a new deep heat source.

At fixed spatial momentum with mode energy $E>0$, $g_E(t)=e^{-iE|t|}/(2E)$ is continuous, with derivatives $-i/2$ and $+i/2$ at zero from opposite sides. Hence its second distributional derivative has a $-i\delta (t)$ term, while away from zero $g_E^{\prime \prime }+E^2{g}_E=0$. Therefore $({\partial }_t^2+E^2)g_E=-i\delta (t)$ and, after the spatial Fourier transform, $(\square +m^2){\Delta }_F=-i{\delta }^4$. This checks the propagator numerator, contour signs and equal-time canonical commutation relations independently of residue calculation.

Electromagnetic radiation consists of propagating oscillations of the electric field and magnetic field that carry energy and momentum.

Thermal electron-positron pairs in chemical equilibrium with photons have opposite chemical potentials when the one-particle energy includes rest energy. For negligible degeneracy and $k_{B}T\ll m_e{c}^2$, the zero-chemical-potential density per charge species is $n_0=2[m_e{k}_{B}T/(2\pi {\hslash }^2){]}^{3/2}{e}^{-m_e{c}^2/(k_{B}T)}$. In that nondegenerate limit, charge neutrality and the classical equilibrium product $n_{-}{n}_{+}=n_0^2$ give $n_{+}=(\sqrt{n_b^2+4n_0^2}-n_b)/2$, with net Electron density $n_b$. Pairs are important when $n_0\gtrsim n_b$. At arbitrary temperature the zero-chemical-potential marker uses the integral $n_0=({\pi }^2{\hslash }^3{)}^{-1}{\int }_0^{\infty }{p}^2[\exp (\sqrt{m_e^2{c}^4+p^2{c}^2}/k_{B}T)+1{]}^{-1}dp$. Its ultrarelativistic limit is $3\zeta (3)(k_{B}T/\hslash c{)}^3/(2{\pi }^2)$. A nonzero net chemical potential shifts the exact abundance boundary; the zero-potential curve is an approximate crossover marker.

An extensive quantity scales in proportion to the size of a thermodynamic system: combining $n$ independent identical copies multiplies it by $n$. Energy, entropy and volume are examples.

The focusing power wave equation is $u_{tt}-\Delta u=u|u{|}^{p-1}$. Its conserved energy has the negative potential term $-|u{|}^{p+1}/(p+1)$ and is not positive definite. Localized ordinary differential equation blowup for a wave equation produces smooth compactly supported data with finite maximal lifespan for suitable focusing powers.

A harmonic map between Riemannian manifolds is a critical point of the Dirichlet energy $\frac{1}{2}\int |dQ{|}^2$. For a Euclidean domain and target unit sphere, the equation is $\Delta Q+|\nabla Q{|}^{2}Q=0$. Stationary wave maps are exactly harmonic maps of their spatial domain.

The healing length of a repulsive homogeneous condensate is the length at which density-gradient kinetic energy and interaction energy balance. The displayed convention sets the dimensionless Laplacian coefficient to one in the stationary Gross–Pitaevskii equation. A hard-wall number density profile has width $\sqrt{2}{\ell }_h$; definitions without the factor $\sqrt{2}$ occur, so the equation and convention must be specified.

A holonomic constraint is a relation $f(q,t)=0$ involving generalized coordinates and possibly time, rather than a genuinely nonintegrable velocity relation. Independent smooth constraints of full row rank define a local configuration submanifold. Ideal constraint forces are normal to its tangent virtual displacements and can be represented by Lagrange multipliers. A time-dependent holonomic constraint need not conserve mechanical energy.

Joule heating converts electrical work into internal energy. A resistor dissipates $P=IV=I^{2}R$; in an ohmic continuum the local rate per volume is $\mathbf{j}\cdot \mathbf{E}=\eta j^2$, with resistivity $\eta$. Wind-driven currents are a proposed source of heating of an inflated giant planet.

Taking $\bar{E}=GMr/q^2>0$ in a radial orbit time transformation gives $r=(\bar{E}/GM)q^2$ and $\bar{\Phi }(q)=-E(\bar{E}/GM{)}^2{q}^2$. Bound Kepler orbits with $E<0$ map to a confining harmonic oscillator; positive energy maps to an inverted oscillator. Full angular reconstruction halves the original orbital angle.

The $n=0$ level of the transverse harmonic oscillator in a uniform magnetic field, with energy $\hslash |qB|/(2m)$ before longitudinal kinetic energy or spin contributions.

A stationary particle of rest mass $m$ absorbing a photon of energy $E$ acquires total invariant massThis follows from four-momentum conservation, since the incoming photon has momentum magnitude $E/c$. If the system splits into two equal particles, the largest kinematically permitted daughter rest mass is $M/2$, attained when both are at rest in the centre-of-momentum frame. Their common laboratory speed at threshold is $cE/(m{c}^2+E)$. Above this daughter-mass threshold there is insufficient centre-of-momentum energy; below it, relative kinetic energy is available.