Search Results (69)

Standard pushers for the relativistic equations of motion of a charged particle in an electromagnetic field—Boris, Vay, Higuera–Cary—do not, in general, preserve the full symplectic structure of the underlying Hamiltonian system, while high-order non-symplectic schemes such as Runge–Kutta accumulate secular error over long times. We propose a two-stage, symmetry-preserving framework (SP-PINN) for the 3+1-dimensional relativistic dynamics of a charged particle in a prescribed field, including a focused Gaussian laser pulse, that pairs a physics-informed neural network with an explicit symplectic integrator: the network learns a surrogate relativistic Hamiltonian, while the integrator—which is not itself learned—advances it. In Stage 1, an unsupervised physics-informed neural network learns the surrogate from the covariant equations of motion using a Lorentz-invariant loss that enforces the mass-shell constraint $\mathcal{H}=m{c}^2\gamma$; in Stage 2, the surrogate is advanced with an explicit symplectic map built on Tao’s extended phase space, valid for the non-separable relativistic Hamiltonian. To isolate the geometric integrator from neural-network approximation error, every benchmark figure advances the analytic relativistic Hamiltonian through Stage 2, the learned Stage-1 surrogate being assessed separately. We benchmark against the Boris pusher and Runge–Kutta on three core test problems (adding the Higuera–Cary pusher in the symplecticity diagnostic), supplemented by plane-wave, ensemble, and pulse-family studies, and we measure the first Poincaré–Cartan loop invariant directly as a quantitative diagnostic of symplecticity. The magnetic-field test illustrates the contrast between bounded and secular error growth: Runge–Kutta drifts secularly, the Boris pusher conserves the invariants to machine precision as a volume-preserving gyro-integrator, and the symplectic map keeps the error bounded for all time; on a non-integrable magnetic trap, where no exact volume-preserving rotation exists, the symplectic map alone keeps the energy error bounded. The learned surrogate is the current accuracy bottleneck—not yet competitive with the conventional pushers for the static cases—but for the demanding laser case, a vector-potential light-cone reformulation reduces this surrogate error to $(3.0\pm 0.1)\times {10}^{-4}$ (three seeds) and yields learned trajectories that remain phase-coherent over essentially the whole interaction. The framework targets laser–plasma acceleration, synchrotron-radiation modeling, and particle tracking. Full article

We study a two degrees of freedom Hamiltonian system describing the motion of a particle in a potential field in the form of a $S^1$ symmetric double well, namely $V=-(x_1^2+x_2^2)+{(x_1^2+x_2^2)}^2$, known also as a champagne bottle potential. This system is completely integrable. The champagne bottle is the simplest member of a class of integrable systems that have no global action variables due to a non-trivial monodromy. Beyond that, the geometric and dynamical properties of the system near the equilibrium are of primary interest. We calculate the Birkhoff normal form and the nontrivial action near the focus–focus singularity and obtain the semi-global symplectic invariant near the focus–focus point, which is introduced by Vũ Ngọc. Examples of such calculations are still few. We compare our result with the semi-global symplectic invariant of the spherical pendulum. Full article

When a symplectic integrator is applied to an integrable Hamiltonian system, the step transition map can be viewed as a nearly integrable symplectic map with a step-size-dependent perturbation. In this paper, we establish a Nekhoroshev-type theorem for such maps, providing explicit exponential stability estimates that apply directly to small-twist cases. We explicitly emphasize that our analysis treats a small but strictly positive twist ($m>0$) with quantified m-dependence, rather than a uniform degenerate-twist result; accordingly, the admissible perturbation threshold explicitly deteriorates as the twist constant m approaches zero. Consequently, we prove that symplectic integrators exhibit exponential stability over exponentially long time scales. Our results, derived via resonant normal form construction and the geometric covering of the action space, furnish rigorous and computable bounds for the long-term preservation of action variables, thereby offering new insights into the nonlinear dynamical behavior of geometric discretizations. Full article

This study evaluates structure-preserving neural network architectures for learning holonomically constrained mechanical dynamics in Cartesian coordinates. In contrast to methods using reduced coordinates, the full ambient phase space ${\mathbb{R}}^{2n}$ is retained with explicit algebraic constraints $C_i\left(\mathbf{q}\right)=0$ to provide a test bed for constraint-aware learning. The Constraint-Aware Hamiltonian Neural Network (CA-HNN) is proposed, which augments the standard HNN with a dedicated multiplier network ${\lambda }_{\varphi }(\mathbf{q},\mathbf{p})$ for Lagrange multipliers and a composite loss function evaluated on predicted rollouts. The theoretical framework is grounded in the geometry of constrained Hamiltonian systems: the extended phase space ${\mathbb{R}}^{2n+m}$ carries a degenerate antisymmetric structure where an m-dimensional kernel encodes constraint directions, while the symplectic structure emerges on the $2(n-m)$-dimensional reduced manifold $\Sigma$. It is proven that the physical Hamiltonian is conserved on the constraint surface under augmented flow. Benchmarks on a pendulum ($C=x^2+y^2-l^2$), double pendulum, and bead on a parabola ($C=y-x^2$) demonstrate that CA-HNN reduces constraint violations $\parallel C\left(\mathbf{q}\right)\parallel$ by $5\times$ to $2400\times$ compared to standard HNNs. While the best energy conservation is achieved by PINNs, these findings clarify the roles of architectural inductive bias, constraint augmentation, and soft physics regularization. Full article

We develop structure-preserving variational integrators for non-autonomous Lagrangian systems by extending the prolongation–collocation variational integrator framework to explicitly time-dependent dynamics. The proposed method is obtained by discretizing Hamilton’s principle for non-autonomous Lagrangians, leading to a family of discrete Lagrangian functions defined at a fixed time step. By combining Hermite interpolation, the Euler–Maclaurin quadrature formula, and collocation applied to the Euler–Lagrange equations and their prolongations, the resulting scheme retains key qualitative properties of variational integrators, including a discrete symplectic (or cosymplectic) structure and favorable long-time behavior. We clarify the relationship between the proposed integrator and classical variational integrators for autonomous systems, showing that the method naturally reduces to the standard prolongation–collocation formulation in the time-independent case. Numerical experiments on representative examples illustrate the effectiveness of the approach and demonstrate its advantages over standard integration methods for non-autonomous systems. Full article

Integral-form nonlocal elasticity provides a mechanically meaningful approach to describing size effects, yet it leads to Volterra-type integro-differential equations that are difficult to solve analytically and numerically challenging for boundary layers and high-order modes. In this work, we developed a symplectic numerical integration framework for Eringen’s two-phase (local/nonlocal mixture) integral model by embedding the constitutive operator into a Hamiltonian formulation and discretizing the influence domain in a belt-wise manner. A step-increase strategy was incorporated to allow flexible spatial marching while preserving the geometric (symplectic) structure of the transfer operation. In addition, a symmetry-explicit, element-level stiffness representation was derived for the discretized integral operator; it exposes a mirrored long-range coupling pattern and enables symmetric, energy-consistent assembly. The resulting kernel-agnostic algorithm accommodates both smooth and finite-range kernels. Static benchmarks and longitudinal vibrations are investigated for exponential, Gaussian, and triangular kernels over representative length ratios and mixture parameters. Comparisons with available analytical and asymptotic solutions show good agreement within their validity ranges, and the method yields stable higher-order eigenfrequencies when asymptotic expansions may be unreliable. The current study is limited to a linear one-dimensional rod setting, and validation is restricted to published analytical/asymptotic solutions rather than experimental calibration. Full article

Eringen’s integral constitutive relation is more general than its differential counterpart for modeling small-scale effects in micro- and nanostructures; however, it leads to integro-differential governing equations that are difficult to solve, which has limited the practical use of integral formulations. To directly address this gap, this paper introduces a novel symplectic-based numerical method that efficiently and accurately analyzes the free vibration of small-scale Kirchhoff plates governed by Eringen’s integral nonlocal model. The method discretizes the nonlocal integral operator by introducing inter-belt elements for long-range interactions and adopting a truncated influence domain, while balancing computational efficiency and accuracy. The effects of the nonlocal parameter, two-phase mixture parameter, mode numbers, kernel types, and geometric parameters on the natural frequencies are systematically investigated. The results indicate stiffness softening. For a simply supported square nanoplate with side length $a=10 \mathrm{n}\mathrm{m}$, the first-order frequency parameter decreases by approximately 25% as the nonlocal parameter increases from 0 to 4 nm, and higher-order modes exhibit substantially greater sensitivity to nonlocal effects. Convergence and accuracy are validated against published continuum-level solutions and molecular dynamics simulations; relative deviations are below 2% in most cases, and the local limit $\left(l_a=0\right)$ yields errors on the order of ${10}^{-3}$. Full article

Integral quantization is a powerful framework for mapping classical phase-space functions—defined on a symplectic manifold—onto quantum operators in a Hilbert space. It encompasses several quantization methods, such as coherent-state quantization, and inherently incorporates operator symmetrization. The formalism relies on a choice of weight function, whose flexibility allows for a family of possible quantizations. In this work, we address the inverse problem: given a quantum operator, how can one determine a classical phase-space function whose integral quantization reproduces exactly that operator? We propose a systematic method, within the integral quantization framework, to construct such a classical function, which depends on the chosen weight. We demonstrate that quantizing the resulting function recovers the original operator, thereby establishing a consistent two-way mapping between classical and quantum descriptions. The method is applied to several physically relevant operators: the projector, a mixed-state density operator, the annihilation operator, and an entangled state. We also analyze how quantum entanglement manifests in the structure of the corresponding classical phase-space function. Full article

The transition of the mining industry towards Industry 5.0 demands predictive models capable of strictly adhering to physical laws and modeling complex, non-Euclidean geometries—capabilities often lacking in standard graph neural networks. This systematic review, conducted under the PRISMA 2020 protocol, analyzes the emergence of “Era 5” architectures by synthesizing 96 high-impact studies from 2019 to 2026, focusing on Clifford (geometric algebra) GNNs, simplicial and cell complex neural networks, symplectic/Hamiltonian GNNs, and generative flow networks (GFlowNets). The analysis demonstrates that Clifford architectures provide superior rotational equivariance for robotic control; Simplicial networks capture high-order topological interactions critical for geomechanics; Symplectic GNNs ensure energy conservation for stable long-term simulation of structural dynamics; and GFlowNets offer a novel paradigm for generative mine planning. We conclude that shifting from data-driven approximations to these mathematically rigorous, structure-preserving architectures is fundamental for developing reliable, physics-informed digital twins that optimize structural integrity and operational efficiency in complex industrial environments. Full article

We present a construction of the anisotropic Gaussian semi-classical Schrödinger propagator, emblematic of a class of Fourier integral operators of quadratic phase kernels related to the Schrödinger equation. We deduce a set of algebraic relations of the variational matrices, solutions of the variational system pertaining to single Gaussian wave packet semi-classical time evolution, some already known in the literature, representing the symplectic and other invariances of the dynamics, which are subsequently utilized in order to derive the Van Vleck formula from the semi-classical Schrödinger propagator. Full article

We present the ContEvol (continuous evolution) formalism, a family of implicit numerical methods which only need to solve linear equations and are almost symplectic. Combining values and derivatives of functions, ContEvol outputs allow users to recover full history and render full distributions. Using the classic harmonic oscillator as a prototype case, we show that ContEvol methods lead to lower-order errors than two commonly used Runge–Kutta methods. Applying first-order ContEvol to simple celestial mechanics problems, we demonstrate that deviation from equation(s) of motion of ContEvol tracks is still 𝒪(h⁵) (h is the step length) by our definition. Numerical experiments with an eccentric elliptical orbit indicate that first-order ContEvol is a viable alternative to classic Runge–Kutta or the symplectic leapfrog integrator. Solving the stationary Schrödinger equation in quantum mechanics, we manifest ability of ContEvol to handle boundary value or eigenvalue problems. Important directions for future work, including mathematical foundations, higher dimensions, and technical improvements, are discussed at the end of this article. Full article

Quantum mechanics postulates wave–particle duality and assigns amplitudes of the form $e^{iS/\hslash }$, yet no existing formulation explains why physical observables depend only on the phase of the action. Here we show that if the quantum of action ${\hslash }_{\mathrm{geom}}$ is finite, the classical action manifold $\mathbb{R}$ becomes compact under the identification $S\equiv S+2\pi {\hslash }_{\mathrm{geom}}$, yielding a $U\left(1\right)$ action space on which only modular action is observable. Wave interference then follows as a geometric necessity: a finite action quantum forces physical amplitudes to live on a circle, while the classical limit arises when the modular spacing $2\pi {\hslash }_{\mathrm{geom}}$ becomes negligible compared with macroscopic actions. We formulate this as a compact-action theorem. Chronon Field Theory (ChFT) provides the physical origin of ${\hslash }_{\mathrm{geom}}$: its causal field ${\Phi }^{\mu }$ carries a quantized symplectic flux $\oint \omega ={\hslash }_{\mathrm{geom}}$, making Planck’s constant a geometric topological invariant rather than an imposed parameter. Within this medium, the Real–Now–Front (RNF) supplies a local reconstruction rule that reproduces the structure of the Feynman path integral, the Schrödinger evolution, the Born rule, and macroscopic definiteness as consequences of geometric compatibility rather than supplemental postulates. Phenomenologically, identifying the electron as the minimal chronon soliton—carrying the fundamental unit of symplectic flux—links its spin, charge, and stability to topological properties of the chronon field, yielding concrete experimental signatures. Thus the compact-action/RNF framework provides a unified geometric origin for quantum interference, measurement, and matter, together with falsifiable predictions of ChFT. Full article

We propose an end-to-end pipeline for Orthogonal Time–Frequency Space (OTFS) sensing that integrates deterministic signal processing with a Machine-Learning (ML) inference stage. The pipeline first generates a complex delay–Doppler grid via standard Symplectic Fast Fourier Transform (SFFT)-based OTFS reception. We then employ an ’oracle’ Ground-Truth (GT) association process to deterministically label signal peaks, extracting their complex gain ($\alpha$) and absolute indices $(m,n)$ to deduce physical targets (range, radial velocity). These oracle-aligned labels are used to train a Random-Forest (RF) classifier. The RF model learns to map normalized $33\times 33$ complex patches, centered on signal peaks, to their corresponding target parameters. On an 80/20 split of 10,000 samples, the classifier achieved a 0.966 accuracy, 0.965 macro-F1 score, and 0.998 macro Receiver Operating Characteristic–Area Under the Curve (ROC–AUC). Notably, when tested on held-out scenes, the model’s derived range and velocity predictions achieved $100\%$ coincidence with the GT, while amplitude and phase corresponded in $89\%$ of instances. This hybrid oracle-and-ML approach demonstrates a highly effective and robust method for precise target extraction in OTFS-based sensing systems. Full article

This paper presents an orthogonal time–frequency space (OTFS)-based integrated sensing and communication (ISAC) framework for vehicular-to-base-station (V2B) scenarios, where a synthetic road environment models vehicular mobility and multipath propagation with explicit ground truth. In the sensing stage, OTFS probing signals with Gray-coded quadrature amplitude modulation (QAM) are processed via inverse symplectic finite Fourier transform (ISFFT) and cyclic prefix orthogonal frequency-division multiplexing (CP-OFDM). The receiver applies cyclic prefix (CP) removal, fast Fourier transform (FFT), and symplectic finite Fourier transform (SFFT) to extract delay–Doppler (DD) responses. Channel estimation uses time–frequency least squares (TF-LS), robust background suppression, constant false alarm rate (CFAR) detection, and non-maximum suppression (NMS), yielding Precision = 0.79, Recall = 0.84, and F1 = 0.82. Communication decoding employs per-bin least squares, minimum mean-squared error (MMSE) equalization, and Gray-mapped QAM demapping. Across ten frames at 20 dB SNR, the system decoded $1.887\times {10}^8$ bits with $1.575\times {10}^5$ errors, producing a bit error rate (BER) of $8.34\times {10}^{-4}$. Error vector magnitude (EVM) analysis reports mean = 0.30%, median = 0.06%, confirming constellation stability. Random Forest (RF) and imbalanced RF (IRF) classifiers trained on augmented DD payloads achieve Precision = 0.94, Recall = 0.87, and F1 = 0.92. Results validate OTFS-ISAC as a robust framework for V2B communication and sensing. Full article

The rapid integration of renewable energy sources into modern power systems has exacerbated power quality challenges, particularly broadband oscillation phenomena that threaten grid symmetry and stability. The proposed symplectic geometric mode decomposition (SGMD) method advances the field; however, issues like mode aliasing and over-decomposition are unresolved within the symplectic geometric paradigm. To resolve these limitations in existing methods, this paper proposes a novel time-frequency-coupled symmetry mode decomposition technique. The approach first applies symplectic symmetry geometric mode in the time domain, then iteratively refines the modes using frequency-domain Local Outlier Factor (LOF) detection to suppress aliasing. Final mode integration employs Dynamic Time Warping (DTW) for optimal alignment, enabling accurate extraction of oscillation characteristics. Comparative evaluations demonstrate that the average error of the amplitude and frequency identification of the proposed method are 1.39% and 0.029%, which are lower than the results of SVD at 5.09% and 0.043%. Full article