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For square plates with a linearly varying gap from $a-\delta$ to $a+\delta$, the local-strip approximation gives the displayed capacitance for $|\delta |<a$. Its expansion is $C=({ϵ}_0{s}^2/a)[1+{\delta }^2/(3a^2)+O({\delta }^4/a^4)]$. An isolated capacitor has fixed electric charge, so $U=Q^2/(2C)$ decreases as $|\delta |$ increases. A sinusoidal small tilt therefore produces a leading energy variation at twice the tilt frequency.

A time-dependent holonomic constraint is a positional restriction $f(q,t)=0$ whose permitted configurations change explicitly with time. Here $q$ denotes the generalized coordinates. A bead confined to a moving wire is an immediate example: its allowed position is specified at each instant, but the wire carries the permitted set of positions through space. The distinction between actual motion and a virtual displacement at fixed time is essential.^[ref]

Differentiating one smooth holonomic constraint along an actual trajectory givesA permitted virtual displacement, however, compares neighboring allowed positions at the same instant:The missing time term is not an approximation. Actual velocity and a tangent virtual displacement obey different conditions when the constraint moves. In the following power calculation, take $q$ to be fixed Cartesian coordinates, so explicit time dependence belongs to the constraint rather than to a moving coordinate system.

For an ideal constraint, its generalized forces may be written $Q_j^{\mathrm{c}}=\lambda \partial f/\partial q_j$, with a Lagrange multiplier $\lambda$. They do zero virtual work, yet their actual constraint power isThus an ideal, frictionless moving constraint can exchange energy with the constrained body. Zero virtual work does not imply zero actual work. The actuator maintaining the imposed motion is part of the physical energy accounting.

For a worked apparatus model, put a bead of mass $m$ on a smooth horizontal wire whose height is prescribed as $z=h(t)$ in a uniform downward gravitational field of magnitude $g$. The wire permits horizontal motion and supplies a vertical normal force $N$. The constraint is $f=z-h(t)=0$, and Newton's second law givesThe bead's mechanical energy, including gravitational potential energy, isbecause horizontal motion is unforced. For $m=0.5\mathrm{kg}$, $g=9.81m/{\mathrm{s}}^2$ and $h(t)=\frac{1}{2}a{t}^2$ with $a=2m/{\mathrm{s}}^2$, the wire exerts $5.905\mathrm{N}$; at $t=1\mathrm{s}$ it delivers $11.81\mathrm{W}$ to the bead. A vertical normal force is perpendicular to every allowed horizontal virtual displacement, but parallel to part of the actual velocity. Measuring wire motion and bead load would expose precisely the energy transfer that is lost by treating the moving wire as a stationary, workless support.

Expanding a state in unperturbed energy eigenstates with their free phases converts the time-dependent Schrodinger equation into coupled equations for slowly varying amplitudes. To first order, the perturbation on the right-hand side acts on the initial unperturbed state.

This is the time-dependent branch of perturbation theory; its corrections describe transition amplitudes.

For a time-independent Hamiltonian operator $H$, an energy eigenstate satisfiesFor a one-dimensional particle of mass $m$ in a real potential energy $U(x)$,Its full time-dependent wavefunction is $\Psi (x,t)=\psi (x)e^{-iEt/\hslash }$: substitution in the Schrödinger equation checks both the minus sign and the factor $\hslash$. Thus an energy eigenstate has stationary position probability density, even though its quantum phase evolves. The boundary conditions and operator domain are part of this eigenvalue problem, selecting physically allowed solutions; the infinite square well and Radial Schrodinger equation for the hydrogen atom give complementary confined examples.

For a complete discrete orthonormal basis of energy eigenstates, linearity gives $\Psi (x,t)={\sum }_n{c}_n{\psi }_n(x)e^{-i{E}_{n}t/\hslash }$, with ${\sum }_n|c_n{|}^2=1$. Cross terms in $|\Psi {|}^2$ oscillate at angular frequencies $(E_n-E_m)/\hslash$. A continuous spectrum requires the corresponding spectral integrals; for a free particle on the whole line, individual plane waves are generalized eigenfunctions, and a normalized Gaussian wave packet is their integral quantum superposition. Adding a spatially constant real $C$ to $U$ shifts every energy eigenvalue by $C$ and multiplies the full state by $e^{-iCt/\hslash }$, a global phase; energy differences, position probabilities and probability current are unchanged.

For a short applied perturbation $V(t)$ in a Quantum Hamiltonian, define $K=\int V(t)dt$. Since $V$ has energy units, $K$ has action units. In the ideal impulsive limit, during which the unperturbed evolution is negligible, the quantum state changes by $\exp (-iK/\hslash )$ if the perturbation commutes with itself at different times. More general pulses require time ordering.

Writing the impulse as $V(t)=K\delta (t-t_0)$ makes the dimensions explicit: the Dirac delta function in time has inverse-time units. For an off-diagonal two-state impulse with coupling $K_{21}=k$, the transition probability is ${\sin }^2(|k|/\hslash )$. The ratio $|k|/\hslash$, rather than $|k|$ alone, is the dimensionless pulse strength. Integrating across the full pulse gives its full area; an artificial factor of one-half must not be introduced by treating the pulse as an endpoint without specifying a regularization.

Time-of-flight mass spectrometry separates ions by measuring how long they take to reach a detector after a timed acceleration pulse. The simplest instrument has an ion source, an accelerating electric field, and a drift tube kept under vacuum. For ions of equal positive electric charge accelerated through the same electric potential difference, smaller mass gives greater speed and earlier arrival.^[ref]

Let $U$ be the accelerating electric potential difference, $q$ the positive electric charge, $m$ the mass, and $L$ the field-free drift length. Neglecting initial speed and the time spent accelerating, conservation of energy givesFor an ideal $1\mathrm{m}$ drift at $1000\mathrm{V}$, a singly charged ion of mass $100\mathrm{u}$ arrives after approximately $22.8\mu \mathrm{s}$; $400\mathrm{u}$ takes $45.5\mu \mathrm{s}$. Here $\mathrm{u}$ is the unified atomic mass unit. Quadrupling mass doubles the ideal flight time. A doubly charged $200\mathrm{u}$ ion has the same mass-to-charge ratio as the singly charged $100\mathrm{u}$ ion and therefore the same ideal arrival time.

Real peaks broaden because ions begin at different positions and with different kinetic energies. In 1955, W. C. Wiley and I. H. McLaren analysed these effects and demonstrated an ion gun with two accelerating regions that reduced timing differences caused by initial position. Comparing calculated focusing conditions with measured spectra turned a timing idea into a practical improvement in mass spectrometry.^[ref]

A time-reversal operator $T$ intertwines forward and backward evolution:It must be antiunitary in ordinary quantum mechanics. Its conjugate linearity sends $i$ to $-i$, allowing it to commute with a time-reversal-invariant Hamiltonian rather than reversing the sign of its energy.

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.

An effective low-energy lattice model obtained from a strongly repulsive Hubbard model near one electron per site. The retained Hilbert space excludes double occupancy. Its usual nearest-neighbor Quantum Hamiltonian has projected hopping $-t{\sum }_{⟨ij⟩,\sigma }({\tilde{c}}_{i\sigma }^{†}{\tilde{c}}_{j\sigma }+h.c.)$ and antiferromagnetic superexchange $J{\sum }_{⟨ij⟩}({\mathbf{S}}_i\cdot {\mathbf{S}}_j-n_i{n}_j/4)$. Virtual intermediate states with one double occupancy cost energy $U$, giving $J=4t^2/U$ to leading order. Three-site hopping terms of the same expansion order are omitted in this simplified version.

At half filling the projected hopping is blocked. Doped holes restore charge motion coupled to spin correlations. Nearest-neighbor singlet pairing can support a d-wave bond order ansatz on a square lattice, but the Quantum Hamiltonian alone is not a proof of a unique superconducting ground state for all parameters.

In static thermal equilibrium, the local temperature times the stationary lapse is constant. Energy measured locally redshifts to conserved Killing energy by that same lapse; requiring a single equilibrium Boltzmann exponent gives the relation. The Killing vector field normalization fixes the constant but not the ratio of temperatures at different locations.

Tomi K. Baikie coauthored research on photosynthesis.^[ref]

Photosynthesis captures light energy and transforms it into chemical forms that a biological cell can use. In plants, algae and cyanobacteria, this supports carbon fixation: incorporation of carbon from carbon dioxide into chemical compounds used to build living material. Capturing light and incorporating carbon are distinct processes coupled through metabolism. See OpenStax Biology 2e, 8.1: overview of photosynthesis.

A topological insulator has an insulating bulk electronic spectrum and boundary electronic behavior constrained by a nontrivial bulk band topology and the relevant protecting symmetries. In common three-dimensional time-reversal-invariant examples, the conducting surface can be approximated at low energy by a Dirac cone. The number of surface cones and the range of the linear approximation depend on the material; boundary conduction and thermal bulk phonons therefore need separate state counts.

A connected component, or homotopy class, of admissible field configurations with fixed asymptotic conditions. Distinct topological sectors cannot be joined by a continuous deformation within that configuration space. A nonzero topological charge labels a sector, but a stable finite-size energy minimum also depends on the energy functional.

Torque measures a force's turning effect about a specified origin. The geometry in the angular momentum diagram applies with force replacing momentum: transverse forces give positive or negative torque, while a radial force gives zero. A longer perpendicular lever arm makes the same force more effective, explaining why pushing near the end of a handle is useful.

For a force $\mathbf{F}$ applied at position $\mathbf{r}$ from a fixed origin $O$ in an inertial frame, its torque, or moment of force, isHere $F=|\mathbf{F}|$, $\varphi$ is the angle between position and force, and $\ell =r\sin \varphi$ is the perpendicular lever arm. Direction follows the right-hand rule; in the $xy$ plane ${\tau }_z=x{F}_y-y{F}_x$, positive counterclockwise as viewed from $+z$. The International System of Units unit is $\mathrm{N}\mathrm{m}=\mathrm{kg}{\mathrm{m}}^2/{\mathrm{s}}^2$. Although torque and energy have the same physical dimensions, torque is an axial vector and energy is a scalar; their roles differ.

For $\mathbf{r}=(2,0,0)\mathrm{m}$ and $\mathbf{F}=(0,3,0)\mathrm{N}$, ${\tau }_z=+6\mathrm{N}\mathrm{m}$. Applying the same force at $(1,0,0)\mathrm{m}$ gives $+3\mathrm{N}\mathrm{m}$; reversing it gives a negative torque; directing it along $\mathbf{r}$ gives zero. More generally, sum ${\mathbf{r}}_i\times {\mathbf{F}}_i$ to find the total torque about one common origin.

For a constant-mass point mass, the product rule and Newton's second law show why net torque changes angular momentum:The first cross product vanishes because its factors are parallel. For a collection with canceling internal torques, the remaining sum is external torque, giving conservation of angular momentum when it vanishes.^[ref]

A nonzero torque can turn the direction of angular momentum as well as change its magnitude. Indeed, the dot product gives $d|{\mathbf{L}}_O{|}^2/dt=2{\mathbf{L}}_O\cdot {\mathbf{\tau }}_O$: a perpendicular torque changes direction without instantaneously changing the magnitude. This is the geometry behind gyroscope precession. For a rapidly spinning symmetric top with a fixed tip, centre of mass distance $d$ from the tip, mass $M$, and spin-dominated angular momentum magnitude $L\simeq I\omega$, the slow steady precession approximation gives $\Omega \simeq Mgd/L$, where $g$ is gravitational acceleration. ^[ref] This is an approximation with $\Omega \ll \omega$; the uniform precession of a heavy symmetric top retains contributions omitted by the spin-dominated estimate.

For axial stretch $\lambda$ and twist per reference length $\alpha$, the deformation in local cylindrical orthonormal bases hasThe principal stretches follow from the eigenvalues of $F^{T}F$. For a cylinder of reference radius $A$, stored energy per reference length is $\mathcal{E}=2\pi {\int }_0^{A}W(F)RdR$. Reversible power $M\dot{\alpha }+N\dot{\lambda }=\dot{\mathcal{E}}$ gives torque $M={\mathcal{E}}_{,\alpha }$ and axial force $N={\mathcal{E}}_{,\lambda }$. The relevant derivative holds the reference twist fixed, not the twist per current length.

For steady rotation at nonzero current, the electromagnetic braking torque is $C{I}^2$ and balances applied torque $G$. Thus $I^2=G/C$, while the circuit requires $\omega =R/C$. With $H=NI/l$, these give the displayed field and $G\omega =I^{2}R$. This is a steady energy balance, not a general nonlinear stability claim.

For a steady Keplerian accretion disk with inward accretion rate $\dot{M}$ and vanishing inner torque, the outward angular-momentum torque is $\mathcal{G}=\dot{M}(\sqrt{GMr}-\sqrt{GM{r}_{\mathrm{in}}})$. It transports energy outward at rate $\Omega \mathcal{G}$. The binding-energy acquisition rate between radii $b$ and $a$ is $B_{a,b}=GM\dot{M}(1/a-1/b)/2$. Integrating the vertical viscous dissipation and then the annulus givesSince $\Omega \mathcal{G}=GM\dot{M}[r^{-1}-\sqrt{r_{\mathrm{in}}}{r}^{-3/2}]$, subtraction proves the displayed energy balance. Far from the inner edge $D_{a,b}\simeq 3B_{a,b}$, whereas the boundary torque powers vanish for the complete disk and $D=B=GM\dot{M}/(2r_{\mathrm{in}})$. Local dissipation is therefore not supplied solely by local changes in orbital binding energy.

An ideal fluid issuing from a small hole a vertical depth $h$ below a free surface has speed

Here $h$ is the vertical distance from the free surface to the centre of the orifice, and $g$ is the positive magnitude of gravitational acceleration. The ideal prediction assumes incompressible flow, the same ambient pressure at the free surface and emerging jet, negligible velocity of the reservoir's free surface, and negligible viscous dissipation and surface tension effects. For a falling free surface, a quasi-steady approximation additionally assumes that the local outflow adjusts much faster than $h$ changes. It applies the Bernoulli equation at each instant, rather than claiming that the whole drainage is steady flow.^[ref]

A short proof uses a streamline connecting the reservoir's free surface at elevation $z_s$ to the emerging jet at elevation $z_o$. With constant mass density $\rho$, negligible surface speed and no mechanical energy losses, the Bernoulli equation givesEqual pressures cancel, leaving $v^2/2=gh$: a unit mass gains the same kinetic energy as a body in free fall through height $h$. This makes Torricelli's law a useful first estimate before the detailed velocity field is known. At one quarter of the original head, ideal jet speed is one half as large; it is not proportional to head.

Far from the accelerating outlet region, the reservoir has approximately hydrostatic pressure $p_{\mathrm{ambient}}+\rho gh$ at the outlet's elevation. The emerging jet instead has approximately ambient pressure when surface tension is negligible. Do not assign that bulk hydrostatic pressure to the emerging jet as well as adding the height difference: that would count the available mechanical energy twice. A sealed reservoir, a submerged outlet or significant surface tension changes the pressure conditions and requires a different balance.

Actual volumetric flow rate is commonly represented by a discharge coefficient:where $a$ is the geometric area of the orifice. This definition compares volumetric flow rates, not just speeds. If a nearly uniform jet at its vena contracta has area $a_j$ and speed $v_j$, define the two separate ratios $C_c=a_j/a$ and $C_v=v_j/\sqrt{2gh}$. Then $Q=a_j{v}_j$ gives $C_d=C_c{C}_v$. A contracted jet can consequently have a speed close to $\sqrt{2gh}$ while $C_d$ is appreciably below one. No particular $C_d$ is universal: calibrate it for the opening and operating range.

Torsional strain is an increase in the energy of a molecule associated with an unfavourable torsion angle, relative to a stated reference conformation. Rotation about a bond can change the relative orientations of substituents without changing which atoms are bonded. The IUPAC definition of strain explicitly includes unfavourable dihedral angles alongside bond-length and bond-angle distortions.^[ref]

Consider ethane viewed along its central carbon–carbon bond. An eclipsed conformation aligns projected carbon–hydrogen bonds, whereas a staggered conformation places them between one another. An idealized threefold potential energy model iswith $\varphi =0$ chosen eclipsed. It gives $V(0)=V_3$ and $V(\pi /3)=0$: a finite rotational barrier separates equivalent minima. The model is a teaching parametrization; $V_3$ must be obtained independently rather than read from the drawing.

The 1937 paper by J. D. Kemp and K. S. Pitzer explicitly connected the entropy of ethane and the third law of thermodynamics with hindered methyl-group rotation.^[ref] This historical connection shows why internal rotation is a measurable molecular degree of freedom, not simply unrestricted motion around a single bond. In a ring, geometric closure can restrict accessible torsion angles; ring strain can therefore include torsional strain as well as other contributions. Assigning a separate torsional energy is model-dependent, rather than a complete account of every electronic interaction.

A circular shaft twisted uniformly through small end angle $\varphi$ over length $L$ has local engineering shear strain $r\varphi /L$ at radius $r$. Its shear stress is $Gr\varphi /L$, so the integrated torque is $C={\int }_{A}r(Gr\varphi /L)dA=G{J}_p\varphi /L$, using the polar second moment of area. The restoring torque has the opposite sign to the imposed twist. The stored elastic energy is $C\varphi /2=G{J}_p{\varphi }^2/(2L)$. The circular cross-section is important: general noncircular sections can warp and have a different torsion constant.