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For $b>0$, put ${\psi }_b(z,t)=\psi (bz,bt)$. Each derivative gains a factor $b$, so ${\mathcal{L}}_b(z,t)=b^2\mathcal{L}(bz,bt)$. Changing integration variables to $(Z,T)=(bz,bt)$ contributes $b^{-2}$, proving exact invariance of the full-space action.

Set $b=1+\eta$. At fixed coordinates, $\delta \psi =\eta X$ with $X=z{\psi }^{\prime }+t\dot{\psi }$. Since the density is quadratic in derivatives,The variation is a divergence, so subtract its boundary term in the Noether current:Thus the dilatation charge of a one-dimensional elastic wave isAn overall sign change would give the same conservation law. Unlike $Q_1$, this charge has units of action, $\mathrm{J}\mathrm{s}$. Its explicit factor of $t$ is balanced by transport of the weighted momentum density; an explicitly time-dependent charge can still be conserved.

For the ideal dilute Maxwell-Boltzmann gas implicit in a single-particle partition function, indistinguishability gives $Z_N=Z_1^N/N!$. With Stirling's approximation,Differentiate at fixed $T,V$:This is a thermodynamic-limit ideal-gas result; arbitrary interacting classical gases do not have $Z_N=Z_1^N/N!$.

Equilibrium constraints and the Gibbs minimum. The open-system equilibrium contacts allow exchange of heat, volume and matter. For unconstrained contacts without external-field or interface corrections, equilibrium requires equal temperatures, pressures and each exchangeable species' chemical potentials across the contacts. These conditions follow by maximizing total entropy under conserved total energy, volume and species amounts.

The next minimization statement fixes the system's particle number. For a system in a reservoir at $T_0,p_0$, with $N$ fixed and only pressure-volume work, the reservoir entropy change givesFor candidate states at the imposed $T=T_0$, $p=p_0$, the expression in parentheses is $G$. Thus the Fixed-particle-number Gibbs minimum selects stable equilibrium at fixed $T,p,N$, with stationary first variation and nonnegative allowed second variation. If particle exchange remains unconstrained, $N$ is not fixed and this minimization statement must be replaced by one incorporating the particle reservoir; it is not a justification for minimizing $G$ over arbitrary $N$.

The simple-compressible-system first law givesFor a homogeneous extensive single-component phase, the Single-component Gibbs Euler relation follows from $U=TS-pV+{\mu }_{\mathrm{ch}}N$, yielding $G={\mu }_{\mathrm{ch}}N$. For a mixture, the corresponding identities are $G={\sum }_i{\mu }_i{N}_i$ and $dG=-SdT+Vdp+{\sum }_i{\mu }_{i}d{N}_i$. Surface effects or other nonextensive contributions require additional terms.^[ref]

Reaction equilibrium. Let $d\xi$ be an allowed reaction extent for the forward process, so $d{N}_A=-d\xi$ and $d{N}_B=2d\xi$. At fixed $T,p$,For an interior equilibrium allowing both reaction directions, the coefficient vanishes:At fixed $T,V$, minimizing $F$ gives the same condition. The reaction formula-unit conservation gives $M=N_A+N_B/2$, not the total number $N_A+N_B$ of gas particles. This conservation law replaces fixed total particle number for the reacting mixture.

Using the stipulated single-particle functions,The number-based dissociation equilibrium ratio is thereforeThis number-based quantity has a factor of volume; the corresponding concentration ratio $n_B^2/n_A=K_N/V$ depends on temperature alone in the stated ideal model. A standard activity-based equilibrium constant would additionally specify reference states. No bond-energy factor is added to the functions given in the paper.

Define the dissociated fraction of original A formula units by $\alpha =N_B/(2M)$. Then $N_A=M(1-\alpha )$ and $N_B=2M\alpha$, soThe dissociation fraction versus conserved density follows because the left side increases monotonically from zero to infinity on $0<\alpha <1$. At fixed $V,T$, increasing $M$ decreases $q$, hence decreases dissociation. Equivalently, at fixed $M,T$ compression raises $M/V$ and favours A. Association produces fewer independent translating particles; at high density the translational-entropy advantage of the two-particle side is smaller, so equilibrium shifts toward the one-particle side.

In the first-order Taylor expansion correction it suffices to use the zeroth-order derivatives. They giveThe delayed product is consequentlyMultiplying by $r$ and substituting into the two transfer equations producesThe opposite signs of $f$ and $g$ are also required by conservation law of $S+I+R$: this correction changes the timing of the transfer between $S$ and $I$, while leaving the instantaneous death term unchanged.

The law of mass action makes each elementary reaction rate proportional to the product of its reactant concentrations, with the stoichiometric change determining its contribution. Let $A=A_0$ be maintained constant, and let $D_X,D_Y>0$ be the diffusion coefficients. The first reaction consumes two $X$ molecules, the second creates one net $X$, the third consumes one each of $X,Y$, and the fourth consumes $Y$ while creating $X$. Fick's laws for diffusion giveHere the standard deterministic convention takes the event rate of the first reaction as $k_1{X}^2$, so its loss is $2k_1{X}^2$. A convention absorbing a combinatorial factor into $k_1$ leads to the same nondimensional model after consistently redefining that rate constant. The maintained reactant $A$ makes this an open chemical model; there is no conservation law requiring $A$ to decrease with reaction progress.

A spherical bubble displaces liquid volume $4\pi a^3/3$. Neglect its own weight compared with the liquid buoyancy, as required by the stated gas/liquid approximation. Equating $4\pi a^3{\rho }_{l}g/3$ with the quadratic drag givesRetaining a nonnegligible gas density would add the factor $1-{\rho }_g/{\rho }_l$. Let $V_s=\sqrt{4ag/(3C_D)}>0$, so the upward slip is $u_1-u_2=V_s$.

With constant phase densities, adding the two volume balances gives ${\partial }_x[\alpha u_1+(1-\alpha )u_2]=0$. The closed lower end fixes the total volume flux to zero. Solving it together with the slip relation yieldsIndividual bubbles move upward for $0\le \alpha <1$, while small phase-fraction disturbances move at the characteristic speed $q^{\prime }(\alpha )=V_s(1-2\alpha )$. The latter is negative when $\alpha >1/2$: wave speed and bubble speed are different quantities.

A cleared region starts at the closed bottom, with ${\alpha }_L=0$ and ${\alpha }_R=3/4$. Its Rankine-Hugoniot condition speed is $[q]/[\alpha ]=V_s/4$. The intended clearing solution is thereforeIt is an entropy shock for the concave conservation law flux, since $q^{\prime }(0)=V_s>V_s/4>q^{\prime }(3/4)=-V_s/2$. Its speed also equals the bubble speed on its uncleared side, so it follows the last layer of bubbles. Under this continuation the last bubble reaches the outlet whenafter which $\alpha =0$ throughout the pipe.

There is a boundary qualification for this dense initial mixture. The characteristics in the uncleared state point into the pipe from the top. To continue the displayed solution all the way to the stated clearance time, its incoming top trace must remain $\alpha (1^{-},t)=3/4$ until clearing. Initial characteristics alone cover the shock only until $t=4/(3V_s)$, when its backward right-state characteristic first meets the top boundary. Atmospheric pressure by itself is not a boundary condition for this scalar phase-fraction equation. Thus the displayed formula and printed clearance time give the intended maintained-trace branch; a different top phase-fraction closure could change the later evolution. This is the incoming boundary condition in dense bubble clearing, and is separate from the local derivation of the conservation law.

Physics investigates matter, motion, and interactions by building mathematical models and testing their consequences against observations. A physical law supplies a relation; a model specifies the objects, conditions, and approximations used to apply it. Classical mechanics, electromagnetism, thermodynamics, and quantum mechanics organize different aspects of these investigations. Their usefulness comes from turning measurements into predictions that can be checked and refined.

Consider a simple pendulum: how does its oscillation period change when its length changes? For small amplitude, the ideal model gives $T_0=2\pi \sqrt{\ell /g}$, where $g$ is gravitational acceleration. Taking $g=9.81m/{\mathrm{s}}^2$ as the model parameter, a one-metre pendulum has predicted period $2.0061\mathrm{s}$. Increasing its length to four metres gives $4.0121\mathrm{s}$: quadrupling the length doubles the period. The point mass cancels from the equation, so changing it alone does not change this model's answer.

The prediction identifies a mechanism and a way to challenge it. The small-angle approximation replaces $\sin \theta$ by $\theta$; the full ideal equation predicts a longer period at large amplitude. At $60^{\circ }$, its period is about $1.07318T_0$, already a $7.3\%$ change. Measurements at such amplitudes can reveal the limit of the approximation, while appreciable drag requires a damped pendulum. These distinctions are developed in simple pendulum and dimensional analysis. The small-amplitude model is also derived in Ling, Sanny and Moebs, University Physics, pendulums.

Several different experiments can become one calculation. Nondimensionalization rewrites the pendulum equation using $\tau =t\sqrt{g/\ell }$, leaving $d^2\theta /d{\tau }^2+\sin \theta =0$. All lengths share that dimensionless equation; the amplitude remains a parameter. Similarly, conservation laws and symmetry reveal features that many models share. This is one source of the subject's beauty: a compact relation can organize a large range of phenomena.

The connection to evidence remains essential when a model is elegant. In his introductory lectures, Richard Feynman emphasizes testing proposed relations by experiment and using idealizations whose accuracy depends on the conditions. ^[ref] A discrepancy can motivate a better measurement, a better approximation, or a different physical model; the mathematical calculation and the empirical assessment are both part of the explanation.

The product of numbers $a_1,\dots ,a_n$ is the result of their iterated multiplication. The empty product is $1$, the multiplicative identity. For positive factors, $\log ({\prod }_i{a}_i)={\sum }_i\log a_i$, turning a multiplicative conservation law into an additive derivative identity.

When the only nonpotential force points radially, its curl has no radial component. The radial absolute vorticity therefore obeys a local flux conservation law, although vortex stretching can change it along a fluid trajectory.

The Rankine-Hugoniot condition expresses conservation laws across a moving discontinuity of a weak solution. Across a discontinuity $x=\sigma t$ in $u_t+f(u{)}_x=0$, with left and right states $u_{-}$ and $u_{+}$, conservation requiresWriting $[u]=u_{+}-u_{-}$ and $[f(u)]=f(u_{+})-f(u_{-})$ gives $\sigma [u]=[f(u)]$. The same relation holds componentwise for a system of conservation laws; in compressible flow, conservation of mass, momentum and energy supplies the jump conditions for a shock wave. Integrating the weak solution identity over a thin space-time region crossing the discontinuity yields this balance between the swept-up conserved quantity and the difference of fluxes.

The conservation law for salt in a reservoir: the rate of change of its salt content equals inflowing minus outflowing salt transport and any explicit salt sources. Freshwater addition changes salinity without directly adding salt. For a well-mixed fixed-volume reservoir, salt content is proportional to its volume times salinity.

A similarity profile is the time-independent function left after expressing a evolving field in its scaled coordinate, for example $\rho (z,t)={\rho }_{0}f(z/Z(t))$. Integrating such profiles converts a global conservation law into an algebraic relation among the time-dependent scales.

Invariance under a shift of the time coordinate, with other required background structures held fixed. A continuous time-translation symmetry gives an energy conservation law through Noether theorem. A generic expanding cosmological background is not stationary and need not supply a corresponding globally conserved total energy.

A turbulence closure approximates the unclosed higher-order moments in evolution equations for lower-order turbulent statistics. For example, an equation for the velocity correlation tensor involves triple correlations, whose evolution involves fourth-order correlations and pressure. A closure must be checked against conservation laws, realizability and boundary conditions; matching an energy spectrum alone does not establish all these properties.

A viscous shock is a smooth travelling transition produced when positive diffusion resolves a shock of a conservation law. For $u_t+q(u{)}_x=ϵ[D(u)u_x{]}_x$ with $D>0$, an inner coordinate $X=(x-ct)/ϵ$ gives $D(U)U^{\prime }=q(U)-q(u_{-})-c(U-u_{-})$. The endpoint conditions impose the Rankine-Hugoniot speed. If $q$ is strictly concave, the right side is positive between its roots, so the admissible profile increases from its rear state to its front state. This supplies the entropy selection directly from a regularizing mechanism.