Papers

Lumped-element networks give transparent, passive models of acoustic impedances whose low- and high-frequency limits can be read from the circuit, but a systematic, arbitrarily accurate construction for non-rational impedances is lacking. Continued fractions, one- and two-point Padé approximants and their identity with ladder networks and Cauer synthesis are introduced in tutorial form and applied to two examples: the radiation impedance of a baffled circular piston and the throat impedance of a horn with a piston mouth. The even contraction of the piston's continued fraction has positive coefficients, certified by interval arithmetic for the first 100 pairs, and yields a ladder of positive resistances, masses and compliances; with 14 elements the error is 4e-4 up to ka = 5. Beranek's four-element network is, to about 2%, the order-two two-point Padé approximant. A horn modelled as a cascade of waveguide sections is a continued fraction whose tail is the mouth impedance; the infinite exponential horn is a periodic continued fraction, an exact high-pass ladder. The piston networks are passive at every certified order (up to 202 elements) and are extended without changing existing elements; for a finite horn the error of the mouth network passes one to one to the throat impedance.

This paper presents aeroelastic experiments conducted in the ONERA S3Ch transonic wind tunnel using a supercritical OAT15A airfoil mounted on a rig allowing independent control of pitch and heave degrees of freedom (DOF). The study investigates fluid-structure interaction (FSI) mechanisms and lock-in phenomena between transonic buffet and structural modes. Aerodynamic and structural responses were monitored using steady and unsteady pressure sensors, high-speed stereoscopic Particle Image Velocimetry (PIV), accelerometers, and optical sensors. Four configurations were tested: rigid (0 DOF), pitch-only (1 DOF pitch), heave-only (1 DOF heave), and combined pitch-heave (2 DOF). The test campaign spanned selected Mach numbers, angles of attack, and structural natural frequencies for each configuration. Results show that buffet-induced lock-in occurs abruptly when the Mach number increases, but only when the pitch DOF is enabled, in both the 1-DOF pitch or 2-DOF configuration. These findings establish that pitch-buffet frequency proximity is a necessary condition for lock-in. In the 2-DOF configuration, continuous Mach number excursions revealed a three-way coupling involving the buffet, pitch, and bending modes, with the bending mode modulating the coupled dynamics. This coupling generates large-amplitude limit-cycle oscillations (LCO) within seconds, persisting over a narrow Mach range before buffet cessation.

The practicality of realizing high-directivity sources is studied using a generalized sensitivity metrics. These measures are based on the error model adopted in previous studies on array tolerance, but account for heterogeneous array characteristics in an arbitrary source-expansion basis, making it applicable to the study of continuous sources, ideal uncoupled arrays, and heterogeneous arrays with mutual coupling. The formulation provides the ensemble average of realized directivity under the selected error model, making it a fast predictor of anticipated directivity reduction due to source perturbation. The data required to compute these predictions are straightforward to obtain via simulation or measurement, with a significant simplification when the system can be approximated as lossless. The directivity and the proposed tolerance measures allow for the formulation of optimization problems illustrating the tradeoff between these measures. Several examples demonstrate these tradeoffs, special cases, comparison to previous estimates of maximum directivity, and the accuracy of sensitivity as a predictor of statistical performance over large ensembles of design realizations.

Soon after Maxwell's celebrated synthesis of electrodynamics, the discovery of the electron raised fundamental questions about the nature and structure of elementary particles. The picture of a small, rigid, charged body of finite radius faced the difficulty of introducing non-electromagnetic forces or stresses to prevent the charge from dispersing, an assumption that was increasingly regarded as unsatisfactory. The alternative description in terms of a point charge was no more satisfactory, because its electromagnetic self-energy diverges. From the early twentieth century onward, several authors therefore explored modifications of Maxwell's theory, with nonlinear electrodynamics emerging as a particularly natural and radical possibility. In this paper, we revisit at the undergraduate level some of the nonlinear theories proposed in this context and explore their ability to address the classical self-energy problem of charged particles.

We investigate signal transmission in a one-dimensional chain of $n$ locally coupled plasma oscillators with asymmetric bidirectional coupling $(\epsilon_1,\epsilon_2)$. Our results reveal a clear contrast between symmetric and asymmetric interactions: while symmetric coupling suppresses signal propagation, coupling asymmetry enables efficient nonlinear transmission across the network. In particular, sustained long range transmission is achieved when the forward coupling strength exceeds the backward coupling strength ($\epsilon_1 > \epsilon_2$), as demonstrated through a comprehensive phase diagram in the $(\epsilon_1,\epsilon_2)$ parameter space. Furthermore, under asymmetric diffusive coupling, signal transmission can persist even when the first few oscillators in the chain undergo oscillation death. A central finding of this study is the intrinsic generation of higher harmonics induced by nonlinear interactions, leading to a pronounced reversal of vibrational resonance. This nonlinear redistribution of energy facilitates selective amplification and robust propagation of specific frequency components.

A relation between the quantities now called the scalar potential and the vector potential, of the same form, up to a sign, as the one the literature attributes to Ludvig Lorenz in 1867, was written ten years earlier by Gustav Kirchhoff, with the same status: derived, unnumbered and unnamed. This paper documents, on the basis of the 1857 originals, that Kirchhoff obtains the relation as the result of a calculation and does not impose it; that the quantity now called the vector potential bears no name at all in his text, whereas the scalar one bears the name "Potentialfunction", potential function, in both papers; that Lorenz writes those same letters, $\Omega$, U, V and W, and inherits with them the absence of a name for the three components, although the relation he writes is in other quantities, $\alpha$, $\beta$ and $\gamma$, and carries the opposite sign; and that the English translation of one of the two papers suppresses the word "function" from the term "Potentialfunction", with which Kirchhoff names his scalar quantity. The paper concludes that the pattern documented in the 1867 paper predates Lorenz and comes from the source from which he starts: a relation without number or name, components without a denomination, and a vocabulary added later.

How can human researchers use artificial intelligence (AI) to make progress on difficult mechanics problems while retaining control of physical significance, mathematical rigor, and validation? We explore this opportunity through a quantitative stability problem rooted in the Eshelby conjecture. For a uniform dilatational eigenstrain in a homogeneous isotropic solid, we establish a nonlinear two-sided relation between whole-interior strain heterogeneity and a geometry-only distance from the ellipsoid family, with half-order Sobolev scaling that captures morphological length scale. We then develop a complementary finite-shape theory: ellipsoidal subtraction yields an exact residual-field budget from which systematically convergent upper and lower bounds are obtained for a prescribed geometry. Applied to protruded, indented, and necked inclusions, the bounds agree with independent displacement-based finite-element calculations. The study also documents a human-led, AI-assisted research process in which researchers select consequential problems, shape targets, audit arguments, require independent checks, and interpret mechanics, while AI broadens analytical exploration and supports computation. This experience also led us to develop Open Problems in Mechanics, a versioned community-facing map that makes selected mechanics questions, their known/open boundaries, and criteria for progress explicit.

Precise shaping of surface acoustic waves (SAWs) on anisotropic piezoelectric substrates is complicated by elastic anisotropy and electromechanical coupling. Here we introduce a continuous diffractive acoustic lens (CDAL), in which weak phase-velocity perturbations produced by subwavelength metallization, together with diffraction, drive lateral field redistribution. An anisotropic thin-element propagation model enables efficient inverse design of the CDAL contour. Laser Doppler vibrometry verifies prescribed lateral field profiles and multichannel beam splitting with unequal widths. By combining continuous diffractive modulation with discrete source-phase encoding, we further shape the lateral field into a Bessel profile, demonstrating joint amplitude-and-phase control. This work provides a high-precision and fabrication-compatible mechanism for on-chip SAW field modulation.

We identify all radial interaction potentials that generalize Newton's exterior shell theorem, for which a uniformly distributed spherical shell is indistinguishable from a point-source up to strength renormalization and radial rescaling. We call these shape-preserving shell interactions, extending Gurzadyan's cosmological theorem which considers strength renormalization alone. We further expand the theory from monopolar to multipolar sources and establish the corresponding classification in arbitrary spatial dimensions. We find potential families beyond the spherical property subclass and show that the classifications for multipolar shape-preserving interactions form a nested hierarchy, with each lower-rank class being a subset of the next higher-rank class.

A rigorous macroscopic analysis of classical thermodynamics reveals an overlooked contradiction at the foundations of ideal gas theory: the traditional volume dependent formula for entropy violates the fundamental axiom of thermal equilibrium during reversible isothermal processes. This paper resolves this inconsistency by establishing a strict operational distinction between strictly isothermal and quasi isothermal processes. We show that in order for the perfect gas to be strictly consistent with the laws of classical thermodynamics, the thermodynamic entropy of a perfect gas must be a function of temperature only. This independence of volume is also confirmed by a new analysis of the adiabatic free expansion of an ideal gas, which is found to be intrinsically reversible. To reconcile these results with standard thermodynamic practice, we introduce the concept of a pseudo isothermal process (a finite sequence of small adiabatic and isochoric steps) and show that the classical equations for work and heat apply rigorously to these real processes, as opposed to strictly isothermal ones. We conclude by discussing the profound implications of these findings for statistical mechanics, showing that while the corrected thermodynamic entropy is in perfect agreement with Maxwell's kinetic theory, it is essentially different from the Planck Boltzmann formula. Thus, thermodynamic entropy and statistical microstate counting are two different physical quantities, and the present analysis provides a simple way to free classical thermodynamics from the paradoxes of microscopic disorder.

Resolving the anomalous conductivity exponent of the symmetric $\beta$-Fermi–Pasta–Ulam–Tsingou chain by molecular dynamics can require very large systems because of long finite-size crossovers and thermal-contact resistance. Motivated by this computational challenge, we derive a long-wavelength continuum description and investigate whether the kinetic-theory scaling $\kappa\propto L^{2/5}$ becomes accessible with a moderate number of numerical degrees of freedom. The nonlinear elastic field retains the cubic stress of the microscopic interaction and exchanges heat with Langevin reservoirs. A flux-conservative spatial discretization constructs the force and energy current from the same stress, providing a consistent interior transport estimator. For $L>8$, corresponding to approximately $10^3$ mesh nodes and above at the reference resolution, the fitted exponent is $0.399\pm0.004$. Mesh refinement supports the stability of this exponent between the two finer resolutions, although the conductivity amplitude remains resolution-dependent. The main result is thus an accessible finite-size regime near $2/5$, rather than convergence of the absolute conductivity as the mesh spacing vanishes. This formulation provides a practical route to studying anomalous transport in anharmonic systems and a conservative continuum framework for investigating energy flow in nonlinear elastic media.

Many machine learning algorithms rely on the Fourier transform, which is also the inner core of many analog computing devices and, specifically, of optical computing. The square of the modulus of the Fourier spectrum naturally maps onto pairwise interactions, as in the renowned Ising model, making it applicable to combinatorial optimization. Here we show how the nonlinear Fourier transform, related to integrable nonlinear partial differential equations, significantly widens the computational capability and also enables a wide variety of computing gates. We study specifically the case of the nonlinear Schrödinger equation; we introduce an input composed by “tokens”, i.e., batches of information encoded in a multi-step potential, and we show how the nonlinear spectral computing arising in the wave propagation not only generalizes combinatorial optimization to higher order problems, but also realizes boolean gates. In the weakly nonlinear regime, nonlinearity generates higher-order interactions as perturbative corrections to the pairwise terms. In the solitonic regime, the discrete nonlinear spectrum performs digital computation through soliton generation. By using four input tokens, we demonstrate the programmability of all 256 Boolean functions of three variables, including 3-SAT instances. Our results show that integrable nonlinear propagation provides an unprecedented single physical platform for higher-order and logical computation, opening the way to a rich variety of new algorithms for analog computing, machine learning, and classical and quantum information processing.

We show that the familiar Cherenkov burst from a finite superluminal charge track can be understood as a pair of interfering electromagnetic memory transitions. Endpoint accelerations of the track produce opposite memory steps in the transverse vector potential, such that the corresponding finite-frequency radiation is universally fixed, below the endpoint-acceleration-scale, by the memory amplitude and the time for which the temporary memory offset persists. As the Cherenkov direction is approached, the memory amplitudes grow while their separation in observer time shrinks, until the two vector-potential steps overlap and their divergent contributions cancel into a finite Cherenkov burst. This perspective opens the door to using the Askaryan effect as a setting in which individual macroscopic memory transitions may be resolved, providing a new experimental arena for memory-linked infrared physics.

The paper studies a classical non-relativistic particle moving in $\mathbb{R}^{N}$, $(N\geqslant2)$, in the presence of the Kepler-Coulomb force $\boldsymbol{F}(\boldsymbol{r})=\alpha\boldsymbol{r}/r^{3}$, $(\alpha\neq0)$. It is shown that the extended-configuration-space transformation involving the projective transformation of the radius vector $\boldsymbol{r}\to\boldsymbol{R}=\boldsymbol{r}/(1+\boldsymbol{c}\cdot\boldsymbol{r})$ and the differential time scaling $\mathrm{d}t\to\mathrm{d}T=\mathrm{d}t/(1+\boldsymbol{c}\cdot\boldsymbol{r})^{2}$, where $\boldsymbol{c}$ is a real vector-valued parameter with dimensions of inverse length, constitutes a dynamical quasi-symmetry in the sense that it maps Newton's equation of motion $m(\mathrm{d}^{2}\boldsymbol{r}/\mathrm{d}t^{2})=\alpha\boldsymbol{r}/r^{3}$ into the equation $m(\mathrm{d}^{2}\boldsymbol{R}/\mathrm{d}T^{2})=(\alpha\boldsymbol{R}/R^{3}) \operatorname{sgn}(1-\boldsymbol{c}\cdot\boldsymbol{R})$. This quasi-symmetry is independent of, yet complementary to, the well-known Kepler-Coulomb dynamical symmetry, which is SO($N+1$) for negative energies and SO($N,1$) for positive energies. A detailed analytical and graphical analysis of induced mappings between non-degenerate conic trajectories associated with the original and transformed systems is presented. Mappings between velocity hodographs are also briefly discussed. It is then proved that the conservation of energy in the transformed motion implies the conservation of the Laplace-Runge-Lenz vector in the original dynamics.

Efficiently transferring kinetic energy from a slow, high-force driver to a light payload, with a near-uniform output force, requires a transmission whose mechanical advantage rises along a steeply convex profile. Continuous-contour mechanisms (variable-radius drums, cams) become prohibitive at scale through diameter span, inertia and rim burst. The RopeComb bypasses these limits by synthesising profiles ($G = 0 \to 100+$) from discrete constant-radius sheaves deflecting tension members into staggered spans. Each fold's ratio vanishes at first contact, enabling snatch-free engagement at 10 m/s without clutches or dampers. Because the array ratio is closed-form, bounded least squares matches uniform-force targets to within 0.6-1.5% across mass ratios of 100:1 to 10,000:1 using 8-13 members. To carry large carriage reactions without the overturning moment a single array imposes, two independent arrays flank a central guide and drive a partitioned fixed-ratio stage ($n_1 + n_2 = k$). Sizing array ratios and member counts in the fall-count ratio $n_2 : n_1$ cancels 97.7% of the single-array reaction and 95% of its overturning moment in simulated 1,000:1 launches at $k=7$, reducing compliant peak-to-mean payload force from 1.40 to 1.13. All results are numerical.

Quantum-classical auxiliary-field quantum Monte Carlo (QC-AFQMC) is a promising route to using quantum computers for strongly correlated electronic structure problems, but its scalability is limited by a classical post-processing bottleneck: the measured quantum trial state must be used to evaluate overlaps, force biases, and local energies for many walkers throughout the AFQMC propagation. Existing matchgate-shadow approaches avoid the exponential cost of generic classical shadows, but their post-processing remains expensive for large systems. In this work, we show that particle-number-preserving fermionic shadows, implemented as orbital-rotation shadows, are able to drastically reduce QC-AFQMC post-processing costs. We derive efficient estimators for overlaps and their derivatives, giving $O(N^3)$ scaling for overlaps and force biases and $O(N^4)$ scaling for local energies per shadow and walker and with $O(N)$ Cholesky vector. For fixed normalized target determinants, we also prove that the overlap estimator is unbiased with variance upper bounded by $O(\eta)$, where $\eta$ is the particle number, yielding a sufficient budget of $O(\eta/\epsilon^2)$ ideal independent measurements for additive overlap error $\epsilon$. We combine these estimators with a novel virtual-correlation-energy treatment and a GPU implementation and demonstrate QC-AFQMC using orbital-rotation shadows. Performance projections for $\pi$-conjugated molecules up to pentacene with a (22e, 22o) active space and 356 total orbitals show that the full classical post-processing workflow can be completed in slightly more than two days on 8$\times$A100 graphics processing units (GPUs). These results significantly reduce classical post-processing costs for QC-AFQMC and motivate further improvements in preparation and measurement of the trial state on quantum hardware.

A lease grants one holder exclusive authority over a resource for a fixed term, and its safety property guarantees that at most one holder has the lease at any time. Under special relativity “at any time” has no frame-independent meaning for participants in relative motion. This note first restates the classical safety property with light cones, requiring a holder's expiry to lie in the causal past of any later acquisition by a different holder, an order every observer agrees about. Second, the paper modifies classical PaxosLease: the core change is that the containment and quarantine rules each gain a factor of $k = \sqrt{(1+\beta)/(1-\beta)}$, the relativistic Doppler factor, where $\beta$ is relative speed as a fraction of the speed of light.

We study an equilibrium system, where linear molecules with a rigid charge distribution interact via the mean-field generated by the Coulomb coupling. We focus on the nonlinear regime, which is valid close to external charges immersed in the system. The finite molecular size in this regime leads to a nonlocal response at short distances. First, we show that the mean-field potential is non-monotonic, i.e., like charges can attract each other. Second, we show that the polarization induced by an external charge (and hence its solvation structure) depends on the sign of this charge, violating the effective C-symmetry of the macroscopic electrodynamics. We identify several scenarios of this effect, characterize it with the free energy and entropy differences, and compare it with analogous effects in asymmetric electrolytes described by nonlinear Poisson-Boltzmann equations.

We have theoretically investigated the stochastic switching dynamics of a charged microparticle in a quadrupole trap near the principal parametric resonance at atmospheric pressure. We show that this behavior arises from a balance of thermal noise-activated escape and dissipative return between linear and nonlinear dynamical regimes of the particle motion. The switching follows an effective Gibbs distribution with an effective potential barrier between the regimes and effective system temperature, which are both experimentally accessible. Because the effective barrier depends on particle mass and charge, and since thermal noise reveals rather than obscures the transition, this exponential dependence could provide the foundation to precision noise-aware detection schemes. The system further offers a highly adjustable platform for studying activation dynamics in non-equilibrium parametric systems.

This paper presents a systematic approach to plan and execute a time-efficient sailboat trajectory under the challenging stochastic wind conditions for automated sailboat. Unlike deterministic scenarios, stochastic wind disturbances introduce challenges such as gust unpredictability, directional shifts, and fluctuating apparent wind speeds. These characteristics render traditional planning methods, such as Line-of-Sight (LoS) and waypoint approaches, to be ineffective and not applicable, as they often rely on static or deterministic environmental models. This paper proposed a new path planner and controller based on nonlinear model predictive control method. It is compared with a baseline planner and controller. A Lie-algebraic analysis of the sailboat dynamics is carried out to identify the operating conditions under which the vessel loses first-order control authority in surge, and these conditions are embedded as constraints in the NMPC. Finally, simulation results are provided to validate the proposed framework.

The mechanical wave equation, derived from Newton's laws, takes its simplest form in the rest frame of the medium, and appears to lose that form entirely once expressed in the coordinates of another inertial frame — raising the question of whether, and how, it can be written consistently for an arbitrary inertial observer. This puzzle, often glossed over in undergraduate instruction, is typically set aside by invoking the rest frame of the medium as “preferred,” without clarifying what this means for the principle of relativity underlying physical law. We revisit the wave equation from first principles and demonstrate that this apparent non-invariance stems from its expression in Eulerian coordinates. We present two reformulations: a manifestly invariant form using Lagrangian derivatives applied at the level of microscopic particle force laws, and a factorized Eulerian form separating left and right-moving wave solutions. In this context, we explore limitations of Galilean transformation as the low-speed limit of Lorentz transformation, and discuss potential observational connections, particularly regarding wavelength changes in the Doppler effect. Our derivation provides both conceptual clarity and pedagogical value for intermediate and advanced undergraduate physics.

We establish a quantitative stability estimate for codimension-one immersions into a round sphere. Let $(M,g)$ be an oriented Riemannian manifold and suppose that a prescribed shape operator is realised by a smooth isometric immersion $\theta:(M,g)\to\mathbb S^{n+1}$. We prove that, on every relatively compact strongly Lipschitz domain and for every $1<p<\infty$, any Sobolev immersion $\phi$ is close to $\theta$ modulo an ambient rotation, with the $W^{1,p}$-distances between both the immersions and their Gauss maps controlled by the $L^p$ stretching-plus-bending energies of $\phi$. No a priori bounds on the fundamental forms of $\phi$ are required. The proof proceeds by extending the immersions along normal geodesics and reducing the problem to an equidimensional geometric rigidity estimate on the sphere. A finite localisation and patching argument handles the possible non-injectivity of the normal extension of the reference immersion.

In the present study, we explore the existence and characteristics of a rotating vibrational mode in a discrete buckled ring that manifests under transverse harmonic forcing. The study is conducted through mathematical analysis complemented by numerical simulations. The ring is composed of uniformly distributed concentrated masses connected by linear and torsional springs and supported by an elastic foundation. The buckled state of interest results from homogeneous in-plane compression and is characterized by vibrational modes that have a rotating nature because of the rotational symmetry of the system. The resulting vibrational mode, called a rotational soft mode, is characterized by large amplitude rotations of the ring deformation while keeping its angular momentum zero. It is shown that rotational soft modes can be activated via transverse harmonic forcing through parametric resonance, which is possible only when nonlinearities are introduced to the foundation. In the absence of nonlinear behavior, transverse vibrations of the ring cannot lead to excitation of the rotational mode. Analysis of a constrained kinematics model shows that linear stiffness of the foundation leads to decoupling of the rotational mode from transverse vibrations. When the foundation is considered piecewise linear, with different stiffness values in compression and in extension, it is shown that three different behaviors are possible for the dynamic response of the forced ring: i) no rotation, ii) a limit cycle within a limited finite rotation angle range, iii) induced rotation between subsequent equilibrium angles. The present study contributes to the understanding of the activation and control of soft modes in buckled structures and may lead to actuation applications in soft robotics, as well as advanced manipulation strategies of molecular systems in the context of nanomechanics.

The Prandtl-Glauert-Lorentz transformation establishes a deep mathematical correspondence between compressible potential flow theory and special relativity. Despite being implicit in the aerodynamic literature since the early twentieth century, this analogy remains largely unrecognised across the two communities and has never been surveyed in a unified manner. This review presents, in a self-contained and pedagogical framework accessible to both communities, the historical, mathematical, and applied dimensions of this correspondence. After tracing the parallel developments that gave rise to compressible aerodynamic theory and special relativity, we establish the formal equivalence between the Lorentz transformation and the Prandtl-Glauert coordinate change, and reinterpret relativistic concepts such as time dilation, the Doppler effect, and the Lorentz factor within the acoustic framework. Thanks to the added mass concept, we finally extend this analogy and show that the whole mathematical framework of special relativity is exactly mirrored by the linearised compressible potential flow theory.

We study a new quasilinear system of balance laws (i.e. conservative first-order PDEs with source terms) for the numerical simulation by a front-capturing approach of viscoelastic athermal multi-material flows with sharp interfaces (like a mixture of immiscible fluids and solids). It naturally extends with an energy balance that is meaningful to stability and thermodynamics. In particular, the unequivocal definition of solutions to Cauchy problems can be guaranteed in some cases, thanks to a symmetric-hyperbolic formulation.

Sabine's foundational theory established the cornerstone of modern architectural acoustics. However, it fails to predict zero reverberation time in perfectly absorptive rooms. Eyring subsequently addressed this issue by proposing a new formula, which Knudsen later extended to incorporate air absorption. Nevertheless, Eyring's underlying approach contains fundamental theoretical contradictions. To resolve these flaws, the author previously introduced macroscopic and microscopic models for a revised reverberation theory; however, their mathematical unification, rigorous derivation from differential equations, and the mechanism behind Eyring's underestimation remained unclarified. This paper presents a comprehensive generalization and mathematical foundation of the revised theory. The macroscopic model is derived directly from fundamental energy differential equations, incorporating air absorption and a generalized mean residual free path. By reformulating the microscopic model as a sequential convolution process, its fundamental consistency with the macroscopic model is mathematically demonstrated. Crucially, it is shown that Eyring's underestimation can be interpreted as stemming universally from the structural omission of temporal variance expansion ({\sigma}_{n}^{2}=n{\sigma}^{2}) inherent to multiple reflections, regardless of the assumed probability distribution. Finally, ray-tracing simulations validate the scale-invariant accuracy of the proposed theory. Ultimately, this mathematically consistent framework establishes the theoretical limit for future generalized theories in non-diffuse sound fields.

We study a charged particle moving outside a spiral (screw) dislocation in a uniform magnetic field. Although the geometry is locally Euclidean, its global screw identification couples the conserved axial and azimuthal quantities through the combination \(\mathcal{J}=\ell-\beta k\) After eliminating the cyclic coordinates, the radial motion is governed by an inverse-square term determined by $\mathcal{J}$ and a harmonic confining term generated by the magnetic field. The corresponding reduced quantum problem is exactly solvable for $q\mathcal{B}_0\neq0$, yielding an equally spaced spectrum with an orientation-dependent energy shift. The radial probability density depends on $\mathcal{J}$, and opposite values of the screw-modified quantum number have identical spatial profiles but different energies for a fixed magnetic-field orientation. The radial size scales as \(\sqrt{\langle r^2\rangle_n}\propto |q\mathcal{B}_0|^{-1/2}\), with the magnetic length setting the localization scale. At $q\mathcal{B}_0=0$, the confining term vanishes, and the problem reduces to a scale-free inverse-square system. Thus, the screw parameter enters the dynamics through the global coupling of the conserved axial and azimuthal quantities, whereas the magnetic field provides radial confinement and its orientation determines the spectral asymmetry.

Substituting a velocity constraint into the kinetic energy before forming Lagrange's equations yields, in general, incorrect equations of motion. Recent literature calls this failure "Hamel's paradox," a term introduced in 2010. We show that there is no paradox. The inadmissibility of the substitution was recognized in the 1890s; C. Neumann (1899) named the operation the illegitimate form of the kinetic energy, and Hadamard (1895), Chaplygin (1897), Appell (1899) and Hamel (1904) established its limits of validity. Hamel's textbook of 1949, the nominal source of the paradox, states the prohibition and uses Neumann's term; the result remained in print in the specialist literature and in geometric mechanics. The modern debate cites none of this record; we collect it from the primary sources and measure the debate's claims against it. A geometric analysis identifies the substituted function as the restriction of the mass metric to the constraint distribution. If the distribution is integrable, the restriction is the induced metric of an integral manifold, and the substitution is legitimate. If not, the correctly embedded Boltzmann–Hamel equations differ from those of the restricted energy by a single residue, $\mathrm{Res}_\alpha=c^{s}_{\alpha\beta} \omega^\beta P_s$, relative to the complement of the constraint distribution that the shortcut selects. The residue vanishes exactly when the symmetric part of its coefficient vanishes. The two classical admissibility mechanisms, decoupling of the mass metric and vanishing bracket components, are sufficient for this and, for a single constraint, also necessary; for several constraints they are not, which corrects a necessity claim in Hamel's paper of 1904.

We investigate the applicability of Lagrangian and Hamiltonian Neural Network models to a dissipative system that has explicit time dependence in its Lagrangian, Hamiltonian, and total energy. To do so we consider these neural network models for simulated systems of a harmonic one-dimensional, one-component oscillator with damping, as well as without damping for comparison. We find that both the Lagrangian and Hamiltonian approaches are able to predict the empirical physical behavior of the damped oscillator systems and to effectively “learn” to varying degrees the underlying Lagrangians and Hamiltonians, as has previously been shown to be the case with undamped oscillator systems. These investigations elucidate important properties of Lagrangian and Hamiltonian mechanics, including properties that are not manifest when considering systems without explicit time dependence.

Wide-angle beam steering with high directivity is fundamentally limited by diffraction, restricting the scan performance of phased-array systems. Here, we establish the bounds governing beam steering with high directivity in phased-array–lens systems and show that nonlocal dielectric metalenses can operate close to these limits. Unlike conventional metalenses that suppress mutual coupling, our approach exploits nonlocal electromagnetic interactions to laterally redistribute energy and expand the spatial width of phase coherence across the aperture. This enables efficient wavefront engineering for uniformly excited planar phased arrays using conventional linear progressive phasing. We further reveal the roles of diffraction and evanescent waves in determining the maximum coherent-aperture expansion and directivity enhancement. The resulting angular-channel bounds identify two distinct operating regimes for passive linear phased-array–lens systems: scan-resolution enhancement and directivity enhancement with preserved scan range. Guided by these limits, we derive geometry-independent design equations and develop an adjoint-based inverse-design framework for passive linear metalenses, enabling planar and cylindrical implementations that operate close to the fundamental limits. We experimentally validate a compact cylindrical metalens that preserves the $50^\circ$ scan range and scan resolution of a phased array with a $3.5\lambda_0$-wide aperture while providing a $2.5$–$3.5\,\mathrm{dB}$ increase in peak directivity. These results establish nonlocal dielectric metalenses as a practical platform for diffraction-limited wavefront engineering in phased arrays.