Use the conserved $p_{\varphi }=p_{\psi }=I_3\omega$ in the canonical-momentum equation. For a nonzero tilt,Cancelling with the half-angle identities givesIts regular limit at the north pole is $C/2$, although the azimuth itself is not a uniquely defined coordinate at that pole.

At the initially vertical configuration, the two conserved momenta coincide:The apparent $0/0$ angular term at $\theta =0$ is interpreted by its regular limit with these equal momenta. Substitution into energy conservation, followed by $1-\cos \theta =2{\sin }^2(\theta /2)$, givesWriting $C=I_3\omega /I_1$ yieldsThe zero of the potential at the vertical configuration includes the initially specified transverse kinetic energy separately as ${\omega }_0^2$.

Differentiation givesThe angles $\varphi$ and $\psi$ are cyclic, so their canonical momenta are conserved. The Lagrangian is autonomous, so the total energy is also conserved. The momentum $p_{\theta }$ is not generally constant because gravity and the angular kinetic terms depend on $\theta$.

Since $p_y=0$ initially and remains constant, the motion stays in a plane of constant $y$. Setting $P=\sqrt{2m{H}_0}$ and $p=p_{z0}$ givesThe allowed interval obeys $|p+a{x}^2|\le P$. The sign of $\dot{x}$ is set by the initial direction and changes at a turning point. The specified $x_0$ fixes the starting position on the energy orbit, but the data do not specify which sign of the initial transverse velocity is chosen.

The Legendre transform cancels the term linear in velocity and givesIt equals the mechanical kinetic energy. The Hamiltonian has no explicit time dependence, so $H$ is conserved. This is also a direct consequence of the magnetic force doing no work: $q\mathbf{v}\cdot (\mathbf{v}\times \mathbf{B})=0$.

Define $a=q{B}_0/(2L)>0$. Differentiating the Lagrangian with respect to each generalized velocity givesThe coordinates $y$ and $z$ are cyclic, so $p_y$ and $p_z$ are conserved. The momentum $p_x$ is generally not conserved: its equation is ${\dot{p}}_x=-2ax\dot{z}$. In particular, conserved canonical $p_z$ is not the same as conserved mechanical $m\dot{z}$, which changes as the particle moves across the magnetic-field gradient.

With zero scalar potential, the nonrelativistic charged-particle Lagrangian in SI units is kinetic energy plus $q\mathbf{A}\cdot \dot{\mathbf{r}}$. ThusThe velocity-dependent magnetic term changes canonical momentum but does not contribute to the mechanical energy.

Only $A_z$ depends on position. Taking the curl,The field reverses across $x=0$. Its divergence is zero, as required for a magnetic field; the vanishing field on the midplane is important for the crossing and separatrix trajectories below.

Use a conventional car consumption of $6$ litres per $100$ km and the supplied fuel-energy scale of $9$ kWh/litre. Driving $20,000$ km then consumes approximately $1200$ litres or $10,800$ kWh of fuel energy. If an energy-crop route delivers a net usable fuel power of $0.5\mathrm{W}{\mathrm{m}}^{-2}$ averaged over the year, its yield is $4.38\mathrm{kWh}{\mathrm{m}}^{-2}\mathrm{y}{\mathrm{r}}^{-1}$, requiring about $2500{\mathrm{m}}^2$ per car.

For an electric car consuming $0.18$ kWh/km, the annual electrical requirement is $3600$ kWh. With an annual incident solar energy of $1000\mathrm{kWh}{\mathrm{m}}^{-2}$ and module efficiency $20\%$, the module-area estimate isAllowing for spacing, charging and storage losses increases the solar land requirement, perhaps to several tens of square metres, while crop yield and processing losses change the biofuel estimate. Even with such corrections the difference can be many tens to over a hundredfold. The comparison includes the electric drivetrain's lower required input energy rather than incorrectly demanding equal fuel heating value from both routes. Agricultural land competition, biodiversity, fertilizer and indirect land-use change also matter for biofuels; solar infrastructure still needs material and seasonal-system assessment.

Take a fully backed-up peak demand of $60$ GW, an illustrative gas capacity capital cost of £$500$/kW, and an annual capital-recovery factor of $0.10$. The installed cost is £$30$ billion, so the annual capital charge is £$3$ billion. Spread over an illustrative annual electricity consumption of $300$ TWh,A doubled capital-cost assumption doubles this charge. This estimate counts backup capacity even if it runs rarely, which is why capital rather than fuel can dominate its fixed annual cost.

It is not the total cost of a renewable or gas system. Fuel, emissions, maintenance, staffing, gas supply during correlated cold weather, network capacity, startup constraints and reliability standards remain additional considerations. Backup requirements need not equal the sum of intermittent nameplate ratings; demand correlation, capacity credit, interconnection and storage matter. The arithmetic shows that a large reserve power rating can have a modest annualized capital charge per unit of total electricity, while leaving substantial practical and environmental constraints unresolved.

An experience curve models unit cost as $C(Q)=C_0(Q/Q_0{)}^{-b}$, where $Q$ is cumulative production. A doubling reduces cost by a fraction $1-2^{-b}$. For an illustrative twenty-percent reduction per doubling, $b=-{\log }_2(0.8)\simeq 0.322$, so four doublings would reduce the fitted cost to $0.8^4\simeq 0.41$ of its initial value.

Photovoltaic modules are manufactured in large numbers with repeated designs, so manufacturing yield, scale, automation and material use can all produce observable learning. Wind turbines also benefit from repeated manufacture, but installation, foundations, access, grid connection and site quality form a larger project-specific share; moving into poorer or more difficult sites may offset improvements. Nuclear reactors have fewer repetitions, long project times, changing designs and regulation, and substantial site and financing costs. A cumulative-capacity curve can therefore hide rather than explain changes in nuclear project cost.

A experience curve is an empirical conditional relationship, not a physical law guaranteeing success. Input prices, policy, financing, saturation of a manufacturing improvement and the definition of system boundary all affect extrapolation. Declining cost may encourage deployment, which then increases the cumulative-production variable, so correlation alone does not establish a one-way causal effect. Forecasts should test multiple learning rates and distinguish module or turbine cost from the cost of a reliable delivered electricity system.

A battery stores electrochemical free energy. Representative pack specific energy is of order $0.1$-$0.25$ kWh/kg, with electrical round-trip efficiency roughly $85$-$95\%$. It responds rapidly and can be modular, but energy capacity requires more active material, and ageing depends on time, temperature and cycling.

Pumped storage stores gravitational potential energy. At a $400$ m head with $89\%$ generation efficiency, useful energy is $\eta gh\simeq 0.97$ Wh/kg of water. Its low mass-specific energy is acceptable when inexpensive water and large reservoirs are available. Storage volume and generator rating can be chosen substantially independently, making multi-hour service practical, but suitable sites and civil works are major constraints.

Hydrogen stores chemical energy, with a lower heating value around $33$ kWh/kg of hydrogen before allowing for tank mass. Electrolysis, compression or other storage processing, and conversion back to electricity can reduce electrical round-trip efficiency to roughly $25$-$45\%$, depending on the route. Separate tanks or caverns can make long-duration energy capacity less expensive than adding electrochemical cells, but containment, infrastructure, leakage and conversion equipment matter.

The appropriate comparison includes power rating, energy duration, cycle count, system mass or area, and whether heat or electricity is the final useful product. A large fuel heating value does not imply a high electricity-storage efficiency.

Compare average delivered electrical power per land area with consistent boundaries. Illustrative values are about $1000\mathrm{W}{\mathrm{m}}^{-2}$ for the compact plant site of nuclear generation, $5$-$15\mathrm{W}{\mathrm{m}}^{-2}$ for a UK ground-mounted photovoltaic site, $1$-$3\mathrm{W}{\mathrm{m}}^{-2}$ across a wind-farm lease area, around $0.5\mathrm{W}{\mathrm{m}}^{-2}$ for an energy crop after conversion, and perhaps $5\mathrm{W}{\mathrm{m}}^{-2}$ for a reservoir hydroelectric scheme. These are scale estimates, not universal technology constants.

At representative values $1000,10,2,0.5,5\mathrm{W}{\mathrm{m}}^{-2}$, one average gigawatt requires approximately $1,100,500,2000,200\mathrm{k}{\mathrm{m}}^2$ respectively. Biomass therefore has substantial land competition even when fuel can be stored. Solar development competes less with farming if placed on roofs or already developed land. Wind's large separation area is not entirely excluded from agriculture; its actual foundations and roads occupy a much smaller area. Reservoir impacts can be extensive and strongly site dependent.

A nuclear-site number excludes mining, cooling constraints, transport and exclusion areas; a wind lease number includes much land still usable for other purposes. Reporting them as identical measures of permanently occupied land would mislead. Deployment decisions need both the average energy yield and the consequences of the particular land boundary, ecology, food production and local acceptance.

Distributed generation changes the shape of infrastructure rather than eliminating it. A $1$ GW power station is large, but supplying a national mean demand of tens of gigawatts from smaller intermittent generators still requires transmission, distribution, power conversion, forecasting, balancing and often storage or dispatchable reserve. Geographic diversity can reduce weather correlations only if power can move between regions.

For scale, delivering $30$ GW through a network is a continuous transfer of $30$ billion joules each second. Even a distributed average installation of $3$ kW per household needs ten million participating households to reach that power, before allowing for capacity factor. Solar roofs can reduce some local flows while export peaks or winter deficits increase others. District heat, offshore wind connections, interconnectors and fuel transport also have economies of scale.

Large centralized assets can introduce single-point failure, slow construction and financing risk; modular systems can be installed incrementally and offer local control. Conversely, network reinforcement and seasonal adequacy cannot be inferred from the size of an individual panel. A defensible conclusion is that an effective mix may combine small generators with large shared infrastructure. The categorical claim that large-scale infrastructure has become obsolete does not follow from the existence of distributed generation.

Fission divides a heavy nucleus into more tightly bound fragments, releasing energy through the increase in binding energy per nucleon. A neutron-induced chain reaction can sustain heat production in a controlled reactor. Fusion combines light nuclei into more tightly bound products and needs sufficiently high temperature and confinement to obtain a useful reaction rate despite Coulomb repulsion. Both ultimately supply heat that must be converted to electricity, unless a different energy-conversion scheme is explicitly introduced.

A typical fission releases about $200$ MeV per event, corresponding to roughly $8\times 10^{13}$ J per kilogram of fully fissioned heavy fuel. The deuterium-tritium fusion energy of $17.6$ MeV per reaction corresponds to about $3.4\times 10^{14}$ J per kilogram of reacting mixture. These are nuclear reaction energies, not electricity per kilogram of mined or engineered fuel. Actual fuel use, thermal efficiency, parasitic power and incomplete burn change plant output.

Fission has substantial operating experience and can provide high-capacity-factor generation, but faces capital, waste, decommissioning, fuel-cycle and safety constraints. Fusion avoids a self-sustaining neutron chain reaction in the fuel, but its systems still face neutron damage, activated materials, tritium management and demanding heat extraction. Demonstrating a hot plasma or a favorable reaction-energy balance is not the same as demonstrating economical net electricity. The useful comparison distinguishes scientific feasibility, engineering availability and whole-system cost.

Two major suppressions are missing from the one-femtometre rectangular estimate. First, the Coulomb barrier extends far beyond nuclear distances and is not rectangular. A semiclassical integral through it produces the Gamow factor $\exp (-2\pi \eta )$, with $\eta ={\alpha }_{\mathrm{fs}}\sqrt{\mu c^2/(2E)}$; at the illustrative thermal energy it is already of order $10^{-11}$. A thermal average additionally weights the high-energy tail and is not the same as evaluating a single thermal energy.

Second, two protons cannot simply form a stable bound diproton. The proton–proton chain starts with a weak-interaction conversion producing a deuteron, a Positron and a neutrino. The weak transition probability supplies a further very large suppression. Thus the quoted tiny fusion probability is not a pure one-dimensional barrier transmission probability with universal energy-independent meaning.

The Sun contains of order $10^{57}$ nucleons in a hot dense core and offers enormously many encounters over astronomical times. A very small reaction probability per encounter can therefore supply its luminosity while consuming only a small fraction of the fuel per unit time. The small weak reaction rate is also crucial for the long stellar lifetime; a larger core temperature increases burning and triggers expansion and cooling.