The large circle contributes $O((\log R{)}^n/R)$ and the small circle contributes $O(ϵ|\log ϵ{|}^n)$, so both disappear. The enclosed simple poles are $i$ and $-i$, with logarithms $i\pi /2$ and $3i\pi /2$. Their residues are $(i\pi /2{)}^n/(2i)$ and $-(3i\pi /2{)}^n/(2i)$. The positive orientation of the keyhole therefore givesThe upper bank runs outward and the lower bank inward. Their logarithms differ by $2\pi i$; keeping that orientation is essential for the sign.

A branch point obstructs a single-valued analytic continuation around a closed circuit. Continuing $\log z=\log |z|+i\theta$ once around zero increases its value by $2\pi i$. A branch cut removes a curve joining branch points or extending to infinity so that a continuous argument can be chosen on the remainder. Zero and infinity are logarithmic branch points; the positive-axis cut used here selects $0<\arg z<2\pi$. The cut is a choice of representation, not a line of poles of the multivalued logarithm.

Along a level curve, $V_{x}dx+V_{y}dy=0$, with $V_x=V_0\theta /\pi$ and $V_y=V_0(1+\log r)/\pi$. ThereforeIn the intended upper half-plane $0<\theta <\pi$, the slope is positive for $0<r<e^{-1}$, vertical at $r=e^{-1}$, and negative for $r>e^{-1}$. At the positive real boundary $\theta =0$ the slope is zero except at the stationary point $(e^{-1},0)$, where both derivatives vanish and the quotient is indeterminate. More generally, outside the chosen half-plane its sign is the sign of $-\theta (1+\log r)$, with the argument branch held fixed. A convenient parametric description of the curves to sketch is $V\pi /V_0=r(\sin \theta \log r+\theta \cos \theta )$.

For the upper half-plane use $0<\theta <\pi$, and interpret the source's arctangent as this argument. Then $V=(V_0/\pi )\mathrm{Im}(z\mathrm{Log}z)$ is harmonic, so $\mathbf{E}=-\nabla V$ has zero curl and divergence where there is no charge. On the positive boundary $\theta \to 0$ and $y\log r\to 0$; on the negative boundary $\theta \to \pi$. Hence the two boundary values are respectively zero and $V_{0}x$.

Branch qualification: the principal real arctangent alone gives zero on both sides of $y=0$, contradicting the stated negative-axis boundary condition, for example at $x=-1$. The continuous upper-half-plane argument repairs this notation; it is not an optional additive constant.

Choose a continuous argument $\theta$ on a simply connected domain avoiding zero, and put $r=\sqrt{x^2+y^2}$. With $u=x\log r-y\theta$, direct differentiation givesHere ${\theta }_x=-y/r^2$, ${\theta }_y=x/r^2$; these identities specify the branch needed in the written arctangent. Integrating the Cauchy-Riemann equations givesThe literal principal $\arctan (y/x)$ works only separately on suitable half-planes. It cannot stand for a single continuous argument across the imaginary axis.

Complex differentiability gives $u_x=v_y$, $u_y=-v_x$. Differentiating once more, for continuously differentiable first derivatives with the needed second derivatives, yields $u_{xx}+u_{yy}=v_{yx}-v_{xy}=0$ and $v_{xx}+v_{yy}=-u_{yx}+u_{xy}=0$. Thus both components satisfy Laplace's equation. The result is local on the domain of holomorphy; it makes no assertion across a branch cut.

Lossless Hamiltonian focusing can reduce a beam's transverse position width at the expense of momentum spread, but cannot reduce its occupied canonical phase volume. Coupling to the longitudinal direction can redistribute projected emittance without reducing total emittance. True cooling requires exporting entropy to additional degrees of freedom or discarding particles: stochastic feedback, electron cooling or radiative damping coupled to an external environment provide possible mechanisms, with very different efficiency for high-energy protons. Collimation shrinks the accepted distribution at the cost of intensity. Acceleration also reduces angular divergence at fixed normalized emittance, an apparent geometrical cooling rather than a violation of Liouville's theorem. The apparatus and radiation cannot be omitted from the accounting when the bunch is no longer an isolated Hamiltonian subsystem.

The oscillator flow isIts determinant is ${\cos }^2\omega t+{\sin }^2\omega t=1$, explicitly preserving area. The orbits are energy ellipses; stretching one coordinate during transport is compensated by the conjugate direction.

Phase space combines all independent canonical coordinates and momenta. Its Hamiltonian velocity has divergence ${\sum }_i[{\partial }_{q_i}{H}_{p_i}-{\partial }_{p_i}{H}_{q_i}]=0$ by equality of mixed derivatives. Thus an infinitesimal material phase volume has zero fractional rate of change. This is Liouville's theorem, valid also for explicitly time-dependent Hamiltonians. It preserves fine-grained volume, not necessarily the area of any projected coordinate pair.

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.