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Vertical hydrostatic balance gives $p=\rho g(h-z)$; depth-independent horizontal momentum and depth-integrated incompressibility yield the two shallow-water equations. Curling momentum and combining with mass conservation gives $D[(\zeta +f)/h]/Dt=0$. Dot momentum with $\rho hu$ and use continuity to obtain $E_t+\nabla \cdot J=0$ with $E=\rho h|u{|}^2/2+\rho g{h}^2/2$ and $J=u(E+\rho g{h}^2/2)$. It is kinetic plus gravitational potential energy. A no-normal-flow boundary kills the flux and conserves its integral; a localized nonuniform height has excess positive energy relative to the uniform rest state with the same mass, so it cannot decay to that state without dissipation.

Rotational invariance gives $\{m_y,H\}=\{m_z,H\}=0$. Direct calculation givesTake $G=m_x^2+m_y^2+m_z^2$; angular-momentum brackets imply $\{G,m_x\}=0$, and central symmetry gives $\{G,H\}=0$. Hence $(H,G,m_x)$ are in involution.

At $\mathbf{x}=(1,0,0)$, $m_y=-p_z$, $m_z=p_y$. The differentials of $G$ and $m_x$ have independent components in the $p_y,p_z$ and transverse position directions when $p_y,p_z\ne 0$; $dH$ supplies an independent radial/momentum component. Thus they are locally independent and the system is Liouville integrable.

Insert $\rho ={ϵ}_0\nabla \cdot E$ and $J={\mu }_0^{-1}\nabla \times B-{ϵ}_0{E}_t$ into the Lorentz force and use vector identities plus Faraday's law. Rearrangement gives ${\partial }_t{g}_i+{\partial }_j{\sigma }_{ij}=-f_i$. Here $g={ϵ}_{0}E\times B$ is field momentum density and $\sigma$ is the Maxwell stress tensor, whose surface traction transports momentum.

The three jumps express conservation of mass flux, normal and tangential momentum flux, and total enthalpy (energy) flux across the stationary shock.

Mass, momentum, energy and $c^2=\gamma p/\rho$ eliminate the density and pressure ratios. The result isso a supersonic upstream flow becomes subsonic.

Photon absorption removes a photon and transfers its energy and momentum to matter. It contributes a negative photon-number collision source. It differs from Compton scattering, which redistributes a photon's energy and direction while preserving the count. The integrated source can vanish if photon emission and absorption balance.

In the local rest frame of a gas, the pressure tensor is the directional momentum flux $P_{ij}=\int p_i{v}_{j}f(\mathbf{p})d^{3}p$, where $f$ is the phase-space distribution function per unit momentum volume and $\mathbf{v}$ is particle velocity. For an isotropic distribution, $P_{ij}=P{\delta }_{ij}$, recovering the kinetic pressure of an isotropic gas. An anisotropic distribution can exert different normal pressures along different directions.

For fixed channel width $b(z)=\beta z$, cross-sectional area is $A=\beta h^2/2$. With reduced gravity $G\varphi$, uniform cross-sectional fields and downward settling velocity magnitude $w$, volume conservation, momentum and particle deposition flux giveThe hydrostatic pressure force contains ${\int }_0^h(h-z)\beta zdz=\beta h^3/6$. No streamwise widening term occurs in a prismatic channel. Deposition uses the projected boundary width $\beta h$ under the vertical-settling model.

The conjugate momentum to a periodic internal soliton angle generates shifts of that angle. Single-valued wavefunctions, or Bohr-Sommerfeld quantization with zero Maslov index, quantize this canonical momentum. A theta term can shift the relation between canonical momentum and mechanical charge without changing integer canonical labels.

A quantum time-reversal operator is an antiunitary operator reversing spatial momentum and angular momentum. For a time-reversal-invariant Hamiltonian operator, its conjugation of $i$ makes it intertwine forward and backward unitary time evolution without reversing the energy spectrum. A scalar momentum eigenstate may transform as $\hat{T}|\mathbf{p}⟩=e^{i\chi (\mathbf{p})}|-\mathbf{p}⟩$. With unit-modulus phases and a complete normalized basis, expansion of arbitrary states proves antiunitarity.

For a condensate with uniform bulk complex field one, its convergent localized momentum isThe subtraction removes the uniform-background phase-gradient contribution. If $\psi -1=O(r^{-1})$ and $\nabla \psi =O(r^{-2})$, its integrand is $O(r^{-3})$, integrable in two dimensions. Under decaying variations its functional derivative is $\delta p_j/\delta {\psi }^{\ast }=-i{\partial }_j\psi$; the subtraction is a boundary correction and does not change the bulk equation.

For a body of varying mass $M$ ejecting material at fixed relative velocity, conservation of momentum gives a differential relation between $dM$ and the body's change $dv$. Integrating it produces a logarithm of the mass ratio.

A smooth-cutoff scalar propagator is the inverse of a positive regulated quadratic kernel. It agrees with the unregulated scalar propagator at low momentum and decreases rapidly at high momentum. Smooth suppression is not identical to vanishing support. Its cutoff derivative is the line weight in an exact renormalization-group flow.

For ${\varphi }_K(x-X)=c\tanh (c(x-X))$ in the unit kinetic normalization, $\int ({\varphi }_K^{\prime }{)}^{2}dx=M=4c^3/3$. Substitution of $X(t)$ therefore gives the free-particle collective-coordinate effective Lagrangian $-M+M{\dot{X}}^2/2$. Its quantization has momentum $p$ and energy $M+p^2/(2M)$ to this order. The exact uniformly moving classical kink is a Lorentz boost of the static one, with energy $M/\sqrt{1-v^2}$.

At a nonzero Euclidean momentum $k$, the transverse projector of a vector field is $P_T^{ab}={\delta }^{ab}-k^a{k}^b/k^2$. In a Lorentzian Minkowski metric, its mixed-index form is $(P_T{)}^a{}_b={\delta }_b^a-p^a{p}_b/p^2$, defined for $p^2\ne 0$. It obeys $P_{T}p=0$ and $P_T^2=P_T$. A nonzero null four-momentum is insufficient for this particular decomposition because the denominator vanishes.

A two-sided line plume whose axial velocity and density anomaly both have triangular transverse profiles. For half-width $b$, ${\int }_{-b}^{b}f(x/b{)}^{n}dx=2b/(n+1)$. These profile moments set the different coefficients of stored mass, momentum, and their fluxes. It differs from a wall plume because both edges entrain ambient fluid.

A fermionic mode with canonical anticommutation relations $\{b,b^{†}\}=1$ has the symmetrically ordered Hamiltonian operator $H=\frac{E}{2}(b^{†}b-b{b}^{†})=E(b^{†}b-1/2)$. Its empty-state zero-point contribution is $-E/2$. A quantum harmonic oscillator with bosonic canonical commutation relations contributes $+E/2$ instead. For a Dirac field, sum over both particle and antiparticle modes and over spin: there are four independent oscillators per spatial momentum in four spacetime dimensions. Comparing bosonic and fermionic vacuum energies requires a common regularization in quantum field theory, matching degrees of freedom and a stated vacuum-energy subtraction convention.