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Vary $S=\int L(q,\dot{q},t)dt$ with fixed endpoints. Integration by parts gives $\delta S=\int {\sum }_i[L_{q_i}-d{L}_{{\dot{q}}_i}/dt]\delta q_{i}dt$, hence the Euler-Lagrange equations. Define $p_i=L_{{\dot{q}}_i}$ and, assuming an invertible velocity-momentum relation, $H=p\cdot \dot{q}-L$. The equivalent action is $\int (p\cdot \dot{q}-H)dt$. Independent variations of $p$ and $q$ yieldBoundary terms vanish for fixed endpoint coordinates; constraining momenta independently at the same endpoints is unnecessary. Singular Lagrangians require constraint analysis instead of this unconstrained Legendre transform.

For equal masses $m$, the newly conserved relative $y$ energy isIt is kinetic energy relative to the free $y$ center of mass plus the two interactions. Adding its constant center kinetic energy and the already conserved $x,z$ energies gives the total Hamiltonian. The equal-mass oscillator rotations simplify to $J_{ij}=x_i{p}_{x_j}-x_j{p}_{x_i}$, but are not new: for unequal masses their conserved predecessors contain the mass-weight factors. As a concrete check, independent transformations of $X_i=\sqrt{m_i}{x}_i$ rotate an isotropic oscillator at any positive masses. Complete mass equality does not magically make generic $U_{12}(y_1-y_2)+U_{13}(y_1-y_3)+V(z_1-z_2)$ invariant under rotations of physical space.

When $m_1=m_2$, the cosine becomes one and the previously driven $y$ relative subsystem becomes autonomous. Its energy is one new generic independent time-independent integral; equivalently total energy becomes conserved because the remaining subsystem energies already were. Making $m_3$ equal as well introduces no further generic continuous symmetry in the printed model. The arbitrary $U_{13}$ and $V_{12}$ distinguish particle labels and directions, so equal masses alone do not create a permutation or physical-rotation invariance.

A finite “number of conserved quantities” must mean independent generators/integrals: arbitrary functions of any integral give infinitely many. Mass-weighted rotations and free-coordinate symmetries identified in part a already exist at unequal masses. Thus the justified generic increment is one at $m_1=m_2$, and zero further at complete equality. Special potentials or $\lambda =0$ can change the count, but were not stipulated.

The separability of the printed Lagrangian is important. Every $x_i$ is an independent oscillator of the same frequency, irrespective of mass. Define $X_i=\sqrt{m_i}{x}_i$, $P_i=p_{x_i}/\sqrt{m_i}$. Then the complete constant matrix $Q_{ij}=P_i{P}_j+{\omega }^2{X}_i{X}_j$, and angular quantities $J_{ij}=X_i{P}_j-X_j{P}_i$, include each oscillator energy and the rotational symmetries of this mass-weighted oscillator space. Alternatively its explicitly time-dependent constants are $X_i\cos \omega t-P_i\sin \omega t/\omega$ and $X_i\sin \omega t+P_i\cos \omega t/\omega$.

Translations in all $y_i$ give conserved $P_y=\sum p_{y_i}$. Translation of $z_1,z_2$ together gives $P_{z12}=p_{z_1}+p_{z_2}$; $z_3$ is free and gives $p_{z_3}$. Their center-of-mass positions minus the associated constant velocities times $t$ are boost constants. The decoupled relative $z_1-z_2$ motion conserves $E_{z,\mathrm{rel}}=(p_{z_1}/m_1-p_{z_2}/m_2{)}^2{\mu }_{12}/2+V(z_1-z_2)$, where ${\mu }_{12}=m_1{m}_2/(m_1+m_2)$. In mass-weighted versions, the three free center coordinates also have conserved mutual angular momenta. These families are not all functionally independent.

The $y$ relative subsystem alone is driven through $\cos [\lambda (m_1-m_2)t]$. For generic nonzero $\lambda (m_1-m_2)$ its energy, and hence the total energy, is not conserved. Omitting the mass-weighted oscillator symmetries simply because the physical masses differ would miss genuine symmetries of this particular model.

A constant of motion has zero total derivative on every solution, and may explicitly depend on time. For an infinitesimal transformation $\delta q_i=ϵ{\xi }_i$ with $\delta L=ϵdF/dt$, the on-shell first variation is $\delta L=ϵd({\sum }_i{p}_i{\xi }_i)/dt$. Therefore $\boxed{I=\sum _i{p}_i{\xi }_i-F}$ is conserved. Including a time shift $\delta t=ϵ\tau$ gives the corresponding $I={\sum }_i{p}_i{\xi }_i-H\tau -F$. This proves Noether's theorem in its usual point-transformation form. A symmetry of the action up to a boundary term is sufficient; exact invariance of the displayed Lagrangian is stronger than necessary. Discrete transformations by themselves do not give this continuous-charge construction.

For $\varphi =-\Omega$, $p_1=m({\dot{x}}_1-R\Omega \sin \Omega t)$ and $p_2=m({\dot{y}}_1+R\Omega \cos \Omega t)$. The equations are ${\ddot{x}}_1+{\omega }^2{x}_1=R{\Omega }^2\cos \Omega t$, ${\ddot{y}}_1+{\omega }^2{y}_1=R{\Omega }^2\sin \Omega t$. For $\Omega \ne \omega$,At resonance the forced terms are $R\omega t\sin \omega t/2$ and $-R\omega t\cos \omega t/2$. Constants depend on the marble's initial position and velocity. A general cup orientation is obtained by multiplying the vector solution by $\mathcal{R}(-\alpha t)$; the physical inertial solution is unchanged by relabeling the cup axes. Resonant growth eventually invalidates the small-horizontal-motion/neglected-vertical-speed approximation.

The floor contributes gravitational potential $mgz=m{\omega }^2(x_1^2+y_1^2)/2$. Neglecting vertical velocity as specified, for the nonrotating cup orientation $\varphi =-\Omega$ the Lagrangian isFor general $\alpha =\Omega +\varphi$, replace the velocity components by ${\dot{x}}_1-\alpha y_1+R\Omega \sin \varphi t$ and ${\dot{y}}_1+\alpha x_1+R\Omega \cos \varphi t$. The potential remains the same because the paraboloid is axisymmetric.

The cup's rotation relative to the radial basis has angular speed $\varphi$, so its absolute orientation has speed $\alpha =\Omega +\varphi$. With $\mathcal{R}(\theta )$ the planar rotation matrix,At $\varphi =-\Omega$, $\alpha =0$: the cup translates on a circle while its axes remain parallel to fixed inertial axes. Thus $x=R\cos \Omega t+x_1$, $y=R\sin \Omega t+y_1$. Cancellation of orientation rotation does not remove acceleration of the cup's center.

In a frame rotating with angular velocity $\mathbf{\Omega }$, the Coriolis force is ${\mathbf{F}}_C=-2m\mathbf{\Omega }\times {\mathbf{v}}_{\mathrm{rot}}$. In the northern hemisphere horizontal flow is deflected to the right: balanced circulation is counterclockwise about a low-pressure center and clockwise about a high-pressure center. The deflection reverses in the southern hemisphere. Jupiter's counterclockwise southern Great Red Spot is consequently anticyclonic, a high-pressure system. These statements assume a large-scale rotating atmosphere; they do not explain tiny vortices dominated by local boundary conditions.

Fission power combines dense fuel and low direct carbon emissions with high capital commitments, long build schedules, financing risk and expensive decommissioning. Environmental assessment must include mining, enrichment and fabrication, cooling-water demands, normal releases, severe-accident risk and comparison with avoided fossil pollution. None of these is captured by operational carbon emissions alone.

Waste requires containment and isolation appropriate to activity, heat production and decay times. Short-lived operational waste and spent fuel have different handling needs, while geological disposal depends on engineered and natural barriers plus credible institutions. Recycling changes inventories and resource use but does not remove all waste or proliferation concerns. Public consent, transparent regulation, security and intergenerational responsibility are part of the challenge. Quantitative comparisons should specify reactor type, financing and waste policy rather than treating either zero risk or inevitable catastrophe as a physical law.

Storage has multiple performance measures: energy and power density, efficiency, cycle/calendar life, duration, cost, materials, safety and siting. Lithium-ion batteries offer high round-trip efficiency, fast response and compactness, making them useful for vehicles and short-duration grid services. That does not make them optimal for every duty. Pumped hydro can suit large stationary energy stores where geography permits; thermal stores can serve heat directly; chemical fuels or other long-duration systems can trade efficiency for cheap energy capacity. A technology that is excellent per kilogram may be poor for seasonal storage. “Best” requires a specified service and life-cycle comparison.

At fixed electrical output, improving efficiency from ${\eta }_0$ to ${\eta }_1$ reduces fuel and associated combustion emissions by $1-{\eta }_0/{\eta }_1$. A change from 0.33 to 0.45 saves about 27%. If an illustrative large coal fleet emits 3.6 Gt annually, that is nearly 1 Gt, exceeding a hypothetical 0.5 Gt national total and far exceeding a 0.1 Gt electricity-sector total. Thus the comparison can be true on reasonable scale assumptions. It is not established by the words “state of the art”: fleet sizes, existing efficiencies, feasible retrofit fractions and whether UK electricity or all UK energy is meant must be specified. Continued growth and rebound can offset technical savings, and the options are not mutually exclusive.

A constant renewable contribution can come from geothermal resources, some hydro schemes and sustainably supplied biomass; variable resources can also contribute to firm supply through geographic diversity, storage, interconnection and demand management. These examples refute the absolute “never.” They do not show that every renewable mix can economically supply an unchanged constant demand without additional resources. Baseload is a load/system service rather than an intrinsic virtue of one generator. Long low-wind periods and seasonal storage require energy-capacity accounting, while dispatchable clean systems may make an inflexible minimum-output plant less desirable.

For a generous illustrative roof area 100 m${}^2$, annual rain depth one metre and five-metre usable fall, maximum potential energy is $\rho A{h}_{\mathrm{rain}}gH\simeq 4.9$ MJ, only 1.36 kWh per year before conversion losses. Even 25 million roofs give only about 34 GWh annually, around $10^{-4}$ of a 300 TWh electricity system. Tiny intermittent flows and installation embodied energy make the practical result worse. Household roof-rain generation is therefore a very small carbon-saving resource under realistic domestic heights; the small gravitational drop and water flux explain why, without needing to assume an inefficient turbine.

Electricity is a strong early decarbonization target because large stationary sources are comparatively accessible and a low-carbon grid enables cleaner vehicles, heat pumps and industry. For example a heat pump with coefficient of performance three divides electricity-related heat emissions by three per unit useful heat. But grid decarbonization does not automatically outrank every efficiency, methane, land-use or industrial intervention. Marginal abatement cost, cumulative emissions, construction time and available technology determine priorities. A very coal-intensive grid makes electrification alone less effective; a clean grid does not remove aviation or process emissions. The statement is a plausible strategic emphasis, not a deduction valid for every country and timescale.

“Never” cannot be established by one present price comparison. A useful comparison is levelized cost with common financing, capacity and system boundaries. For illustration, solar capital cost 1000 currency units/kW, annual capital-plus-maintenance charge 8%, and capacity factor 0.15 imply about 61 units/MWh. A coal plant with fuel, maintenance and capital costs of the same order is not separated by an immutable physical barrier. These are illustrative inputs, not current price quotations. Carbon costs and air pollution can worsen coal's comparison; intermittency, land and balancing costs can worsen solar's. Compare delivered services as well as panel-level energy. The defensible conclusion is contingent competition, not a universal impossibility.

Hydropower can convert a renewable water flux efficiently, has low operating fuel requirements, and reservoir schemes can provide rapid dispatch, inertia and storage. Long plant lifetimes can make capital expenditure worthwhile, while pumped storage shifts energy rather than producing a new primary resource.

Limitations include finite favorable sites, seasonal drought, sedimentation, habitat fragmentation, blocked fish migration, displaced populations and major civil-engineering/failure risks. Reservoir evaporation and methane emissions can matter, especially in warm flooded landscapes, so life-cycle emissions are not uniformly zero. Run-of-river plants avoid some reservoir impacts but offer less controllable output. A project assessment must compare useful annual energy and firm capacity, ecological and social costs, and the counterfactual electricity system.

A Pelton wheel splits and turns a free jet through nearly 180 degrees in moving buckets, transferring momentum to the runner. For ideal bucket speed $V$, the absolute exit speed is $2V-u$ and force is $2\rho Q(u-V)$. Power $2\rho QV(u-V)$ is maximized at $V=u/2$, giving $P_{\max }=\rho Q{u}^2/2$. If the same inflow must now pass a slit of height $d$, continuity gives $u=Q/d$. Using the preceding flow rate with $H=y_{\min }$,The head must actually be sufficient to supply this jet speed: $u^2/(2g)$ cannot exceed the available hydraulic drop. An arbitrarily small slit cannot produce infinite power from a fixed-height reservoir; it would reduce discharge or raise water until another outlet, including overtopping, operates. The printed scaling presupposes compatible head and fixed-flow conditions.

Neglect upstream speed and hydraulic losses. If $H$ is the upstream free-surface height above the crest, the specific energy at the crest is $H=y+v^2/(2g)$ and flow per unit width is $Q=yv$. At the critical broad-crested control section, extremizing $Q^2=2g{y}^2(H-y)$ gives $y=2H/3$ and $v^2=gy$. ThereforeFinite approach speed, friction and contraction introduce measured discharge coefficients; the ideal result assumes a long enough crest for the near-hydrostatic critical-flow approximation.

Mean kinetic energy $3k_{B}T/2=12.9$ keV means $T=8.6$ keV, not 12.9 keV. For fully ionized D-T with equal electron and ion temperatures, $p=2n_i{k}_{B}T$ gives $n_i\simeq 4.90\times 10^{19}$ m${}^{-3}$. Thusonly about one third of the stated requirement. The specified operating point therefore does not meet that criterion and requires improved confinement/pressure or continuing external heating. If the criterion's $n$ counts all thermal particles instead of ions, the product doubles but remains below its given threshold. A precise fusion gain or duty cycle cannot be inferred from the triple product alone.