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A per-capita quantity is a total quantity divided by the number of individuals in the population.

As a heuristic approximation, inserting $r=6GM/c^2$ into the circular orbit form of Kepler's third law, with $M$ here denoting the total mass, givesThe dominant quadrupole gravitational-wave frequency is twice the orbital frequency, $f_{\mathrm{GW}}=2/T$.

The Tzitzeica equation in light-cone coordinates is $u_{xy}=e^u-e^{-2u}$ and admits a matrix Lax pair.

A Lax pair encodes a nonlinear equation as compatibility of two linear equations depending on an auxiliary spectral parameter.

Commuting mixed derivatives impose an equation on the coefficient matrices of an overdetermined auxiliary system.

For $F=(p^2+W^2{q}^2+aq)/2$, shifting $Q=q+a/(2W^2)$ gives oscillator energy $E=F+a^2/(8W^2)$ and action $I=E/|W|$.

A Hamiltonian that is a sum of one-coordinate quadratic potentials plus linear terms becomes a collection of independent harmonic oscillators after translating each equilibrium position.

Hamiltonians that split into terms $F_i(p_i,q_i)$ have $\{F_i,F_j\}=0$ for $i\ne j$, and their differentials are generically independent because they use disjoint coordinate pairs.

The inverse scattering transform evolves the scattering data of an auxiliary linear operator and reconstructs the potential from the evolved data.

The Korteweg-De Vries equationis an integrable nonlinear dispersive equation. Its sign convention here admits negative solitary waves.

Writing a defocusing nonlinear Schrödinger field as $\psi =\sqrt{\rho }{e}^{iS}$ turns it into a continuity equation for $\rho$ and a velocity equation for $v=S_x$, including a quantum-pressure term.

In a common normalization, the defocusing case is $i{\psi }_t+{\psi }_{xx}-2|\psi {|}^2\psi =0$. The sign distinguishes it from the focusing nonlinear Schrödinger equation.

The focusing and defocusing equationshave Hamiltoniansunder the convention $i{\psi }_t=\delta H_{\pm }/\delta \overline{\psi }$.

The focusing cubic nonlinear Schrödinger equation has the normalizationthe positive cubic term focuses the field and permits bright solitons on zero background.

In light-cone coordinates, the sine-Gordon equation can be written $u_{XT}=\sin u$. Its inverse scattering transform, solitons, breathers, and scaling symmetries make it an integrable system.

The sinh-Gordon equation is an integrable nonlinear wave equation. Under the one-dimensional reduction $u=u(x)$ with $z=x+iy$, it becomesWriting $\varphi =2u$ gives ${\varphi }^{\prime \prime }=4\sinh \varphi$ and the first integral

With normalized $Y_{00}$, the radial ground state $R(r)=C{e}^{-r/a_0}$ hasIf the angular factor is absorbed into the spherically symmetric wavefunction, its normalization constant is instead $(\pi a_0^3{)}^{-1/2}$.

For zero orbital angular momentum and Coulomb potential $-q^2/r$, the radial equation is

Neglecting spin, bound hydrogen eigenstates are labelled $|n,\ell ,m⟩$, where $\ell =0,\dots ,n-1$ and $m=-\ell ,\dots ,\ell$. Their unperturbed energy depends only on $n$.

A no-radial-node Coulomb eigenstate has principal quantum number $n=l+1$ and radial factorIts energy is $-e^2/[8\pi {ϵ}_{0}a(l+1{)}^2]$.

For reduced mass $\mu$, the Coulomb length scale is