Papers

The hydrogen molecular ion and the hydrogen molecule are treated in an approximate calculation based on classical electrodynamics which includes classical zero-point radiation. It is found within the classical theory that a molecular ion is less-well bound than a hydrogen atom plus a distant proton. The smaller binding energy is explained due to the repulsive nature of the force between the proton and the atom when considering the most natural resonant orbit of the electron. The approximate classical electromagnetic calculation gives a binding energy of $2.1eV$ and a proton separation of $0.916A^{o}$ for the hydrogen molecular ion. The hydrogen molecule is formed by adding a single additional electron to the ion. The electrostatic attraction of the electron to the ion gives a binding energy of $4.6eV$ and a inter-proton separation of $0.6A^{o}$ for the hydrogen molecule in this classical electromagnetic approximation.

We describe in detail and provide computer code for constructing optimal geometric braids of three strands from algebraic data encoding the braiding pattern. Our optimality criterion uses the known interpretation of braids as homotopy classes (rel endpoints) of paths in $SL_2(\mathbb{R})$ joining the identity $I$ to some $A \in SL_2(\mathbb{Z})$, i.e., elements of the fundamental group of $SL_2(\mathbb{R})/SL_2(\mathbb{Z})$, a quotient equivalent to the unit tangent bundle of the classical modular surface $\mathbb{H}/PSL_2(\mathbb{Z})$. The main technical result finds the length minimizing geodesic in a prescribed homotopy class. From another perspective, of independent interest, this amounts to shooting the shortest geodesic that connects, with a prescribed number of spins en route, two given unit tangent vectors to the Poincaré (half-)plane $\mathbb{H}$. The length is measured by a Riemannian metric from a family of deformed Sasaki metrics, sometimes called Kaluza-Klein metrics, whereby unit tangent vectors can be interpreted as infinitesimal rotors, called spinners, and the ratio of the mass to the moment of inertia is the deformation parameter. At the universal covering level $\widetilde{SL}_2(\mathbb{R})$, the geometry is one of Thurston's eight model 3D geometries. In the vanishing mass limit, it converges to the better understood Carnot-Carathéodory contact geometry, where our geodesic shooting extends known formulas. The finite mass case is more delicate, requiring numerical solution of a targeting equation. The characterization of the length minimizing geodesics (No-multiplicity Theorem) and the resulting targeting equation are the main original contribution. The exposition is complete and multi-pronged, aimed at a broad spectrum of readers. (Numerous figures are the backbone of the narrative and should be viewed in color.)

Acoustic metamaterials can exhibit unusual effective properties through mechanisms such as energy localization and resonant behavior in their constituent building blocks. Here, we investigate the internal acoustic energy of fluid core-shell spheres and map how resonances emerge from the interplay between material contrasts and shell geometry. The resulting phase diagram reveals an organized resonant landscape across distinct impedance-contrast regimes. From this structure, we obtain two compact predictive relations: one for resonance existence and another for spectral recurrence of successive peaks. Together, they enable the rapid identification of material and geometric combinations associated with targeted resonance responses. The framework is further tested against representative systems from the literature, reproducing their observed behavior. These results provide a physics-based route for screening and tailoring resonant core-shell building blocks before full-wave numerical simulations.

Nano- or micro-particle rotation driven by light has been well known in the fields of optical manipulation and optical physics since the end of the last century. It is viewed as a sort of angular-momentum transfer from light to material, but its microscopic analysis based on the Hamiltonian or the equation of motion has been less developed. We model this rotation with a simple setup of an electrically dipolar or multipolar particle irradiated by circularly polarized laser and comprehensively analyze the Langevin-type equation of motion by using the Floquet theory for dissipative classical systems and the mode separation method. Furthermore, we numerically compute the time evolution of the particle. As a result, we accurately estimate the dependence of the laser-frequency, laser-intensity, particle mass, temperature, and friction (dissipation) on the laser-driven rotation. We determine the “nonequilibrium phase diagram” of the laser-driven rotation in a broad parameter regime, which consists of three regimes: the rotation frequency $\Omega\propto\omega^{-1}$, $\Omega\propto\omega^{-3}$, or $\Omega=\omega$ ($\omega$ is the laser frequency). Comparing our theoretical result with some experiments, we show that the result of the overdamped Langevin equation is qualitatively consistent with the experiments.

Our present investigation into a rather rudimentary problem is motivated by two classes of problems studied since the 70's, cosmological particle creation and its more accessible analog, the dynamical Casimir effect on the one hand, and quantum friction a neutral atom moving along a dielectric surface would experience, on the other. The backreaction effects of produced particles being able to isotropize the expansion of the universe, or to slow down the moving mirror can be understood via the concept of vacuum viscosity arising from fluctuations of the quantum field. We want to track down the origin of this effect by asking the question whether a moving massive $M$ object with a charged internal degrees of freedom $\chi$ interacting with a free unbounded classical field $\phi$ at zero temperature would experience a viscous force, similar to the said precedents. Adopting a microphysics model for optomechanics which can treat the unequal tripartite $\chi$-$\phi$-$M$ interactions, we first perform a nonrelativistic calculation, which seems perfectly legitimate considering the needs of atomic physics, and found the answer to be yes, but a relativistic covariant calculation says no. We identify where the nonrelativistic framework is defective. The resolution of this latent yet real conflict is technically nontrivial but physically quite inspirational. It results in added enriched contents to Newton's first and second laws when the principles of special relativity are enforced, and rules to follow to get the correct nonrelativistic answer

Open-porous materials exhibit pronounced compressibility, nonlinear densification, and power-law scaling of stiffness with density. In this work, we propose a thermodynamically consistent compressible neo-Hookean constitutive model for open-porous solids in which porosity serves as the primary governing variable. The strain-energy density is formulated to couple distortional elasticity of the solid skeleton with a volumetric response governed by deformation-induced porosity evolution, including a bounded representation of pore collapse. The formulation introduces a minimal set of parameters, namely the initial porosity, intrinsic skeleton moduli, and a scalar parameter controlling the onset of densification. A key feature of the model is a modified volumetric term in which the response is normalised by the current porosity, ensuring a physically consistent transition from a porous to a densified state without artificial stiffening. In the small-strain limit, the model recovers classical linear elasticity with effective moduli that may be chosen either from homogenisation bounds, such as the Hashin-Shtrikman estimates, or from Gibson-Ashby-type power-law scaling to capture topology-dependent behaviour. At finite strains, the formulation captures the characteristic nonlinear stiffening and convex stress-stretch response associated with progressive pore collapse. The proposed framework thus provides a compact, flexible, and extensible constitutive description that unifies effective-medium consistency with experimentally observed scaling behaviour, and is well suited for finite element implementation and multiscale modelling of highly compressible open-porous materials. The model is finally validated against available experimental data.

Objects rolling upward on diverging inclined rails exhibit a counterintuitive behavior that has intrigued physicists and students alike. Among the most well-known is the double-cone paradox, where a cone appears to roll uphill due to the geometry of the rails. In this paper, we extend this idea to more general rigid bodies, including spheres and ellipsoids. We examine the physical conditions under which this motion occurs, and we derive the necessary geometric and energetic constraints using analytical mechanics. We then validate our theoretical models through carefully designed experiments.

Extended electrodynamics (EED) leaves the Lorenz gauge condition unimposed and treats the scalar combination $C=\nabla\cdot\mathbf{A}+c^{-2}\partial\Phi/\partial t$ as a dynamical field. For a conserved source with no scalar initial field, $C=0$ and the theory reduces to classical electrodynamics. A source with a nonzero local continuity anomaly has no Maxwell solution, but EED remains well posed and can support a scalar-longitudinal sector. For each radiating frequency of a localized source, the far field separates into the usual transverse Maxwell channel and a scalar-longitudinal channel with a longitudinal electric field, no magnetic field of its own, and a co-propagating $C$ field. Under the field-only energy balance used here, the time-averaged fluxes add without interference. The scalar channel depends only on $\Lambda=\partial\rho/\partial t+\nabla\cdot\mathbf{J}$ and radiates when its moments at $k=\omega/c$ are nonzero. An all-orders multipole formula is derived for this flux. Bound polarization and magnetization sources conserve charge identically and cannot excite the scalar channel. A compensated polarized carrier with a globally neutral anomalous surface layer isolates the channel and gives its dipole flux in closed form. The connection between the adopted flux and a physical stress-energy tensor remains unresolved. These results are conditional predictions of EED and do not imply a failure of charge conservation in classical electrodynamics.

Noise-activated switching between coexisting stable states is a fundamental mechanism underlying stochastic dynamics in systems ranging from chemical reactions to neural networks. While this phenomenon is well understood for stationary attractors, it remains largely unexplored for limit cycles, whose periodic motion cannot be described by a static potential landscape. Here we experimentally demonstrate activated switching between two coexisting limit-cycle attractors in a driven nonlinear system of coupled resonators. Specifically, we introduce controlled fluctuations to directly observe the rare stochastic transitions between two limit cycles and measure their dependence on noise intensity and driving strength. The measured switching rates are well described by a large-deviation theory, which replaces the conventional activation barrier by the action along the most probable transition path. Our results extend the concept of activated dynamics from stationary to limit-cycle attractors and establish a framework for modeling stochastic transitions between limit cycles in driven-dissipative systems.

Since the first gravitational-wave (GW) detection of September 14th 2015 and with hundreds of gravitational-wave sources identified by the LIGO-Virgo-KAGRA network, GW have produced many important results in astrophysics and fundamental physics. Along with planned new data takings, current detectors will be upgraded and new project, such as Einstein Telescope and Cosmic Explorer, are under study. Among noises limiting low frequency sensitivity, vibro-acoustic noises are particularly important. In this work, we focus on the gravity gradient noise (also called Newtonian noise) of acoustic origin, which refers to the small fluctuations in the gravity field resulting from the acoustic pressure field present in the experimental areas of the detector. The induced noise is quantified in an original way, using a detailed nu- merical acoustic model of the experimental room, when the pressure field is excited by the air conditioning system. The method is used for Virgo, but it can be easily extended for future detectors and used to guide the design of caverns and experimental areas.

Radio detection of ultra-high energy particles relies on the propagation and reflection of broadband radio pulses at boundaries between natural media. In the Sommerfeld–Weyl treatment, a spherical wave is decomposed into plane-wave components and their reflection from a single homogeneous interface is calculated exactly. We extend that treatment to an arbitrary number of laterally uniform spherical layers. Both the spherical-wave decomposition of the source and the spherical geometry of the boundary are retained, while the reflection and transmission coefficients of the plane-wave components are replaced by the exact characteristic-matrix coefficients of the layered medium, evaluated at the local incidence angle on the spherical boundary. When the layer contrast is removed, the formalism recovers the single-boundary result to machine precision, and it reproduces the published spherical-surface reflectivity calculation to better than $1.1\%$ at ten HiCal-2 elevation angles, with a mean deviation of $0.6\%$. We reproduce the measured HiCal-1 reflected pulses from their measured direct partners, with a best signed correlation of $0.83$ and a median of $0.70$ across $106$ pairs, of which $101$ show the expected polarity inversion. Applied to the six reported ANITA anomalous-polarity event geometries, the buried-layer refractive index required for a sign change of the reflection coefficient ranges from $1.68$ at the steepest event to $3.8$–$5.4$ at the four near-horizon events. The full waveform calculation gives no non-inverted reflected pulse at any of these angles, showing that shallow, laterally uniform firn layering does not account for the polarity of the anomalous ANITA events. Because the formalism depends only on the complex refractive index of the medium, it applies more generally to isotropic, nonmagnetic stratified media, including ice, lunar regolith, and conducting layers.

For two hundred years - ever since Faraday's first conception of the field lines - hearts and minds of students have been pervaded by erroneous and unsubstantiated claims about these well defined mathematical objects. The most prominent misconceptions include: (1) a notion of the field lines as of 'lines of force'; (2) a notion that the field lines coincide with particle trajectories; (3) a notion that a field line density measures a magnitude of a vector field; (4) a notion that the field lines of a divergence-free field always form closed loops. Even the modern day literature systematically perpetuates some of these claims. We compile here the trivial counterexamples to these claims, providing physics instructors with an efficient and effective way of dispelling these misconceptions.

In thermodynamics, reversible adiabatic expansion can be understood as a limit of stepwise irreversible processes. We make this idea concrete by studying an ideal gas in an insulating cylinder with a frictionless piston supporting a load divided into $N$ blocks. If blocks are removed one at a time, the resulting expansion is a stepwise, irreversible process, but we prove that for a fixed total load, the work done by the gas approaches the reversible limit as the largest block mass tends to zero. On the way to this limit, an interesting work optimization question arises at finite $N$: how does the work done by the gas depend on the order and sizes of the removed blocks? Guided by numerical experiments accessible to advanced undergraduates, we motivate and then prove general answers to these questions. Our main finite-$N$ result proves that for fixed $N$, the optimal stepwise expansion corresponds to a geometric progression of equilibrium pressures, settling a conjecture previously made by Andresen, Berry, Nitzan, and Salamon.

Frequency-domain Huygens' principle and equivalence relations for an infinitely long cylindrical surface enclosing the field sources or scatterers are addressed. Assuming the harmonic dependence of the fields along the cylinder axis, we derive three equivalent versions of line-integral representations for the fields in free space at any point outside the enclosing cylinder. In one of the forms, similarly to the Helmholtz-Kirchhoff scalar diffraction theory, the integrals contain a longitudinal field component and its normal derivative. Another presented form contains longitudinal and normal field components, similarly to the three-dimensional boundary integral representations by Stratton and Chu. This form eliminates the need to calculate the field derivatives. Furthermore, we present a two-dimensional version of the three-dimensional Schelkunoff-Franz representation, which contains only tangential components of the fields, complies with the field equivalence theorem and can be regarded as a rigorous formulation of Huygens' principle for a cylindrical enclosing surface. All three forms of the line-equivalence relation are exact and describe the fields through equivalent field distributions on a line enclosing the cross section of the source region. A physical interpretation of Huygens' principle in terms of conical waves emanated by virtual linear sources on the cylindrical surface enclosing the true sources is given. Specialization of the relations to the intermediate and far-field zones are presented. The derived expressions are applicable for studies of scattering and radiation in a very general class of structures that are infinite and periodic along one direction.

It has been known for a long time that the equation of motion for dissipative linear dynamical systems with constant coefficients cannot be derived from the classical principle of stationary action because the term proportional to velocity in the equation of motion leads to the time derivative of order one half in the Lagrangian. Thus, approaches utilising fractional calculus have been used; however, they suffer from deficiencies both from mathematical and physical points of view. We here present our version of such fractional calculus based approach that provides correct Euler-Lagrange and, ultimately, the Hamilton equations, energy change of the moving body, and an attempt for a geometric interpretation of how energy dissipates.

The most well-known beyond-Fourier heat conduction model, the Maxwell–Cattaneo–Vernotte equation is solved analytically in the presence of heat transfer boundary condition. In contrast to the corresponding Fourier problem, this boundary condition renders the underlying differential operator non-selfadjoint. With a suitable scalar product, the adjoint eigenvalue problem is established. The resulting left and right eigenfunctions constitute a biorthogonal system, allowing the expansion coefficients to be determined from arbitrary square-integrable initial conditions. This enables a convenient infinite-sum analytical solution, which is presented and thoroughly investigated for various values of the model parameters (including near-Fourier and highly hyperbolic regimes) and for two practically important initial conditions (equilibrium and flash pulse initiated). The spectral structure is analyzed in detail, including the occurrence of real, imaginary, and complex-conjugate eigenvalue roots, their asymptotic distribution, and their dependence on the dimensionless relaxation time and Biot number. We find good agreement with corresponding finite-difference numerical solutions. The comparison also illustrates the effects of spectral truncation (i.e., the Gibbs phenomenon) and numerical dissipation near propagating thermal-wave fronts. Completeness of the eigenfunction set is numerically demonstrated. The initial condition induced by the flash pulse is derived analytically. The results provide an analytical benchmark for non-selfadjoint hyperbolic heat conduction, unveil how boundary heat transfer and relaxation time influence the transition between Fourier-like diffusion and thermal-wave propagation, and open the possibility to find and investigate beyond-Fourier heat conduction via heat transfer in experiments and practical applications.

Third-order elastic constants (TOEC) play an important role in nonlinear material characterization, but measurements of TOEC are laborious with large error margins. This Letter presents the equations of wave velocity changes caused by homogeneous temperature variation and uniaxial stress in isotropic media and the expression of TOEC in terms of thermally induced velocity change and thermal strain. TOEC of an aluminum sample were experimentally determined by measuring ultrasonic wave velocity changes in the uniaxial loading test and the thermal modulation test. Experimental results showed good agreement between the two test methods. Owing to the simple test setup and high measurement sensitivity, the thermal modulation test is a potential experimental method to determine TOEC and absolute acoustic nonlinearity parameters.

Gyroscopes, as fundamental inertial sensors, are crucial for rotation measurements in consumer electronics, automotive, and aerospace industries, with the most widely used kind relying on the Coriolis effect. The chip-scale Coriolis vibratory gyroscopes (CVGs) show reduced size, weight, and cost, but remain far lower performance than traditional macroscale CVGs, as the weak intrinsic Coriolis factor sets a fundamental limit on scaling the sensitivity against the inherently louder Brownian noise in microchips compared to the macroscale ones. Here, to overcome this physical limit, for the first time, we propose and experimentally demonstrate the use of third-order singularities lying within cusp catastrophes in the phase-tracked oscillations of an on-chip CVG to facilitate a cubic-root scaling of the Coriolis-effect-induced frequency modulation. Employing this effect, we achieve a three-order-of-magnitude enhancement in the Coriolis factor, yielding a 253-fold improvement in signal-to-noise ratio and a 297-fold increase in precision. Moreover, the cusp singularity enables a previously unattainable ultrasensitive phase-modulated sublinear measurement, achieving a world-record signal-to-noise ratio performance for silicon-chip gyroscopes. These findings not only provide revolutionary advancements in gyroscope technologies, by filling the gap in observing and controlling the singularity-enhanced Coriolis effect, but also shed new light on other ultrasensitive sensing applications.

Friction does work when two objects in contact slide past one another. This process is studied using a simple numerical model consisting of two flexible asperites, one for each object. The force of friction is modeled as a conservative electric force that can be either attractive or repulsive. Because the asperites are flexible, not rigid, the conservative friction force generates thermal energy and can decrease or increase the objects' translational kinetic energies. The concept of pseudowork plays an important role throughout the analysis. The difference between work and pseudowork is equal to the change in internal energy. The numerical model is used to illuminate the reasons why work is typically greater than pseudowork. For a real physical system, this is the source of thermal energy. The processes discussed here fall into two categories: forced sliding and free sliding. With forced sliding, applied forces keep the objects moving with constant velocities. With free sliding, the only force on one of the objects (in the direction of motion) is friction.

The motion of a projectile released from a simple pendulum is analyzed, with particular emphasis on investigating the optimal release angle that maximizes the horizontal range and the corresponding maximum range. This system serves as a simplified model of the Tarzan jump problem. Using simple analytical methods, the optimal release angle is shown to be characterized by a cubic equation and to increase with the initial velocity. In addition, asymptotic expressions for both the optimal angle and maximum range are derived in the limits of low and high initial velocity.

Altermagnets have emerged as a new class of magnetic materials that combine spin-split electronic bands with zero net magnetization. Extending this paradigm to classical-wave systems has, however, been fundamentally challenging because conventional realizations require broken time-reversal symmetry (TRS). Here, we overcome this limitation by introducing two pseudospin degrees of freedom and constructing a pseudo-time-reversal operator that faithfully reproduces the action of its physical counterpart while preserving actual TRS. Building on this framework, we theoretically propose and experimentally realize the first time-reversal-invariant altermagnetic acoustic crystal. Acoustic measurements directly reveal pseudospin-dependent band splitting–a defining hallmark of altermagnetism–under strictly TRS-preserving conditions. Moreover, the altermagnetic acoustic crystal exhibits sublattice-pseudospin locking, enabling flexible control over acoustic pseudospin splitting and filtering. Our work establishes acoustic crystals as a versatile platform for exploring altermagnetic physics and opens new avenues for spin-inspired wave manipulation in nonmagnetic devices.

Cells often estimate chemical concentrations from only a few diffusing molecules, whose stochastic motion limits sensing precision. We consider a perfect monitoring instrument that records the instantaneous molecular density throughout a finite region without perturbing the concentration field or distinguishing molecular identities. The standard Berg-Purcell estimator weights all positions uniformly. Here we derive the optimal spatial weighting within a large class of unbiased estimators for both concentration and gradient sensing. Variance minimization maps to an electrostatic problem. Surprisingly, although the instrument monitors the entire volume, the optimal estimator assigns all weight to its boundary, reducing the uncertainty in both concentration and gradient sensing. We extend the results to arbitrary geometries and spatial dimensions.

In this Brief Communication, we establish the notion of critical damping for generic two-degree-of-freedom system. For given set of masses and stiffnesses, the critical damping corresponds to the real eigenvalue with maximal multiplicity, equal to minus geometrical mean of the eigenfrequencies. This case corresponds to the fastest possible asymptotic decay rate for generic initial conditions. The damping matrix for the critical case is unique up to reflection of one modal coordinate and, generically, non-diagonal. Quite surprisingly, for large difference of the eigenfrequencies, it is also not positive definite. Therefore, physical realization of the critical case will require active elements that provide negative effective damping.

In this study, we investigated the slow dynamic behavior of concrete with varying compressive strengths ($f'_c = 32\sim56$ MPa) and alkali-silica reaction (ASR) damage. Concrete prisms were conditioned by compressive loading and recovery was monitored by coda wave interferometry (CWI). A self-referencing temperature correction technique was used to minimize the effect of ambient temperature changes, demonstrating the importance of careful temperature control and correction. For intact specimens, both the recovery rate, $m_v$, and velocity drop magnitude, $|c|$, generally increased with compressive strength. Recovery times were calculated from the fit parameters, and higher strength specimens were found to recover faster (9.9 h for $f'_c = 32$ MPa, 6.6 h for $f'_c = 56$ MPa). When the same stress was applied, ASR-damaged specimens had increased softening and faster recovery rates, but longer recovery times (up to 458 h). These findings demonstrate the potential of slow dynamics in concrete characterization, through its ability to evaluate the strength of intact samples and its sensitivity to microcracking in damaged samples.

Classical fractons are Hamiltonian systems that can develop attractors after projection onto configuration or shape variables, although the full phase space admits none. We study a scale-invariant, dipole-conserving two-parameter family of fracton Hamiltonians $H_{\alpha,\beta}$. By separating coordinates into scale and shape, we obtain autonomous shape dynamics that admit fixed points which leave a purely scale evolution of the form $R(t)\propto |t|^{\alpha/(\alpha-\beta)}$. The shape fixed points, which determine the distribution of the expanding particles, are central configurations of power-law Riesz potentials. The distinguished model $(\alpha,\beta)=(-2,1)$ is unique: its scale evolution takes the Einstein-de Sitter form $R(t)\propto |t|^{2/3}$, its fixed-point equation is the equal-mass Newtonian central-configuration, its large-$N$ distribution is a homogeneous ball, and its homothetic trajectories admit a zero-energy Newtonian gravitational dual. The fixed points are locally stable, and simulations at moderate $N$ approach them from random initial data. Large $N$ simulations reveal a richer class of fixed-points: bound clusters of approximately fixed physical size retain internal motion, while their centers approach unequal-mass Newtonian central configurations and preserve large-scale homogeneity. A scale-separation conjecture yields an effective unequal-mass fracton dynamics for the centers and a corresponding zero-energy Newtonian gravitational dual. Trajectories generically exhibit a bidirectional arrow of time: scale and shape complexity grow away from a Janus point, while Boltzmann entropy grows logarithmically. Together, these features reproduce the salient structure of a flat matter-dominated cosmology. In the distinguished fracton model, all these cosmological analogues emerge as attractor properties, making it a toy model for cosmological dynamics without fine-tuning.

For more than three centuries, Newton's Second Law (F=ma) has governed mechanics with undisputed success, yet its epistemic authority has rested on empirical adequacy alone. That empirical contingency vanishes under two physical principles - the Principle of Excluded Perpetual Motion (PEPM) and the Weak Equivalence Principle (WEP) - from which the law emerges as a structural necessity of admissible mechanics. Under the PEPM constraint, Suppes' operational measurement protocol grounds gravitational mass and force as independent primitives. The fusion of Stevin's static and Galileo's kinematic inclined-plane analyses establishes F/a as a well-defined, object-intrinsic quantity - the operational definition of inertial mass. The WEP compels the equivalence of inertial and gravitational mass without presupposing Newton's Second Law. Substituting this equivalence into the definition of inertial mass yields the law as a theoretical identity between independently defined quantities. The derivation is anchored in gravity because weight uniquely bridges statics and kinematics, yet the result is independent of the underlying force mechanism. The derivation carries structural consequences: modified-inertia formulations of MOND are inadmissible under the PEPM-WEP constraint; the location-invariance of the Kibble-balance realization of the SI kilogram is secured on first principles; and under the PEPM alone, torsion-balance and free-fall experiments constitute equally direct tests of the WEP. More fundamentally, Newton's Second Law is not an irreducible axiom of mechanics, but a structural consequence of deeper physical principles: energy conservation and the Einstein Equivalence Principle.

We investigate Hamiltonian curl-force systems with indefinite kinetic energy. We first reconsider Berry's polynomial Hamiltonian curl-force model, whose numerically observed closed trajectories motivated an integrability conjecture. A Painlevé analysis yields a non-principal resonance spectrum, so that the corresponding Laurent series cannot accommodate the required number of arbitrary constants of the general solution. The model therefore fails the standard Painlevé test. We then introduce a four-parameter curl-force family and identify the parameter locus on which this system is integrable. We construct a second Hamiltonian, compatible Poisson tensors, separated complex characteristic variables, and a Lax representation. More generally, the separated form yields polynomial integrable curl-force Hamiltonians of arbitrary degree. We also show that the same construction admits a higher time-derivative potentialisation whose free limit is the degenerate Pais-Uhlenbeck oscillator. Finally, we analyse zero-curl invariant reductions and elliptic periodic solutions, and exhibit an isolated periodic orbit outside the integrable regime. This illustrates that closed trajectories alone do not imply Liouville or Painlevé integrability.

We introduce a notion of completeness for two-dimensional second-gradient elastic continua and propose a microstructural route toward its synthesis. A continuum is said to be complete if the Hessian of the stored energy with respect to the second-gradient variable is locally positive definite, so that every nonzero admissible increment of the placement second gradient is quadratically controlled in the highest-order part of the energy about each configuration. Starting from the Principle of Virtual Work, we derive the constitutive relations, equilibrium equations, and admissible boundary interactions for a broad class of fibrous second-gradient continua whose stored energies depend on fiber stretch, stretch gradients, and curvature. The theory is then applied to several continua motivated by pantographic microstructures. Classical pantographic sheets are shown to be incomplete, while bi-pantographic fabrics enlarge the class of components of the second gradient detected by the energy but remain incomplete. Finally, we formulate a tri-pantographic continuum associated with a proposed three-family architecture and prove that it is complete. The examples illustrate how microstructural architecture influences both the completeness properties of an effective continuum and the pointwise form of its higher-order boundary interactions.

We propose a phase field model of a macrocracked single crystal in which the crack and the geometrically necessary dislocations descend from a single energy functional. Energy minimization alone then decides dislocation nucleation, through an integral criterion evaluated in closed form along slip chords. The criterion yields a size effect inaccessible to point-wise strength conditions: a grain-size-dependent yield stress. With slip suppressed the model reproduces Griffith fracture; with fracture suppressed, the nucleation load measured by the full non-smooth solver agrees with the closed-form nucleation criterion to four percent. The coupled computations produce a two-stage response: at loads an order of magnitude below cleavage, dislocation bands emitted from the notch tip blunt and shield it, raising the initiation load; once the crack grows, the bands heal; a compact cluster of like-signed dislocations travels with the tip, its canceling partner walls pinned at the grain boundary, and the dissipated fracture resistance equals the elastic one. In the purely energetic, dissipationless limit, emission shields the crack but does not toughen it; toughening requires dissipation, incorporated in the sequel through the threshold resistance to dislocation motion.

By investigating the origin of the evolutionary discrepancy between the H-function and entropy, we elucidate that the H-function fails to serve as a valid arrow-of-time criterion in the rigorously derived Boltzmann equation for the hard-sphere gas model under the Boltzmann-Grad limit.