Search Results (245)

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

This work analyzes a bi-metric cosmological model where two sectors, characterized by their respective scale factors, interact through a deformed Poisson bracket structure. This deformation is based on the Generalized Uncertainty Principle (GUP). Through a numerical analysis, we study how this interaction affects the expansion dynamics. The results indicate that for positive values of the deformation parameter, the coupling induces an acceleration that leads to a Big Rip singularity in finite time, even in the absence of a cosmological constant. A power-law relation is established between the deformation parameter and the critical time of divergence for the scale factors. Finally, the regime with a negative deformation parameter is also investigated. In this case, the symplectic structure becomes singular, leading to the contraction of one sector and the freezing of the other. Full article

This article discusses the quantization of linear Hamiltonian systems, a historically rich but under explored line of research. The key idea is that a classical linear Hamiltonian system induces on its phase space a compatible complex structure and scalar product, giving rise to a complex Hilbert space where classical dynamics becomes a one-parameter unitary group. Boson Fock quantization of this group then recovers, up to unitary equivalence, the results of canonical quantization. This expository overview traces the development of this framework from foundational works to modern symplectic perspectives, offering a case study in the dialogue between analysis, geometry, and physics. Full article

This paper develops a structured analytical framework and a robust numerical methodology for the spectral stability of time-varying bilateral quaternion differential equations of the form $\dot{q}=A\left(t\right)q+qB\left(t\right)$. By systematically extending classical real matrix theory to non-commutative dynamical systems via exact isometric real representations, this study utilizes the Kronecker product of real adjoint matrices to rigorously elucidate the underlying tensor structure of the bilateral evolution operator. This tensor-based reformulation proves that the Floquet multipliers of the bilaterally coupled system can be strictly decoupled into the product of the spectra corresponding to the left and right unilateral subsystems. Second, a “Scalar-Vector Stability Separation Principle” based on logarithmic norms is proposed, demonstrating that the transient energy evolution of the system is governed exclusively by the Hermitian real parts of the coefficient matrices, remaining entirely independent of the anti-Hermitian imaginary parts (rotation terms). Furthermore, for constant-coefficient and slowly varying systems, the Riesz projection from holomorphic functional calculus is introduced to establish algebraic criteria for exponential dichotomies, thereby revealing a cubic scaling law that relates the robustness threshold to the spectral gap (${\epsilon }_0\propto {\beta }^3$). Numerically, a Quaternion Chebyshev Spectral Collocation Method (Q-CSCM) is embedded within this exact vectorization framework to ensure that the algebraic symmetries of the bilateral system are strictly preserved through the isomorphic mapping. By explicitly constructing the fully discrete Kronecker product matrix via the exact real vectorization isomorphism, discrete energy estimates are utilized to rigorously prove that the numerical scheme successfully inherits the intrinsic spectral accuracy of the Chebyshev approximation. Comprehensive numerical experiments demonstrate that, within the low-dimensional regime, this methodology exhibits substantial temporal approximation efficiency advantages and superior numerical robustness compared to an alternative Legendre spectral baseline, as well as traditional explicit and state-of-the-art implicit symplectic Runge–Kutta methods, particularly when solving stiff and critically stable problems such as nonlinear Riccati oscillators. Full article

The framework of the research whose part of results are published in this work is the category of real vector bundles over finite dimensional differentiable manifolds. The objects of studies are gauge structures on these vector bundles. We are interested in the dynamical properties of the holonomy groups of Koszul connections as well as on their topological properties, i.e., properties that are of homological nature. For the most part the context is the subcategory of Lie algebroids. In addition to other investigations, three open problems are studied in detail. (P1-Affine Geometry): When is a Koszul connection an affine connection? (P2-Riemannian Geometry): When is a Koszul connection a metric connection? (P3-Fedosov Geometry): When is a Koszul connection a symplectic connection? In the category of tangent Lie algebroids our homological approach leads to deep relations of our homological ingredients with the open problem of how to produce labeled foliations the most studied of which are Riemannian foliations. On a Lie algebroid we define two families of differential equations, the family of differential Hessian equations and the family of differential gauge equations. The solutions of these differential equations are implemented to construct homological ingredients which are key tools for our studies of open problems we are concerned with. We introduce Koszul Homological Series. This notion is a machine for converting obstructions whose nature is vector space into obstructions whose nature is homological class. We define the property of Degeneracy and the property Nondegeneracy of Koszul homological Series. The property of Degeneracy is implemented to solve problems (P1), (P2), and (P3). In the abundant literature on Riemannian foliations, we have only cited references directly related to the open problems which are studied using the tools which are introduced in this work. Thus, the property of nondegeneracy is implemented to give a complete solution of the problem posed by E. Ghys, (P4-Differential Topology): How does one produce Riemannian foliations? See our Theorems 12 and 13, which are fruits of a happy conjunction between gauge geometry and differential topology. 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

In this paper, we clarify several issues concerning the abstract geometrical formulation of thermodynamics on non-compact symmetric spaces $\mathrm{U}/\mathrm{H}$ that are the mathematical model of hidden layers in the new paradigm of Cartan Neural Networks. We introduce a clear-cut distinction between the generalized thermodynamics associated with Integrable Dynamical Systems and the challenging proposal of Gibbs probability distributions on $\mathrm{U}/\mathrm{H}$ provided by generalized thermodynamics à la Souriau. Our main result is the proof that $\mathrm{U}/\mathrm{H}$.s supporting such Gibbs distributions are only the Kähler ones. Furthermore, for the latter, we solve the problem of determining the space of temperatures, namely, of Lie algebra elements for which the partition function converges. The space of generalized temperatures is the orbit under the adjoint action of $\mathrm{U}$ of a positivity domain in the Cartan subalgebra $C_c\subset \mathbb{H}$ of the maximal compact subalgebra $\mathbb{H}\subset \mathbb{U}$. We illustrate how our explicit constructions for the Poincaré and Siegel planes might be extended to the whole class of Calabi–Vesentini manifolds utilizing Paint Group symmetry. Furthermore, we claim that Rao’s, Chentsov’s, and Amari’s Information Geometry and the thermodynamical geometry of Ruppeiner and Lychagin are the very same thing. In particular, we provide an explicit study of thermodynamical geometry for the Poincaré plane. The key feature of the Gibbs probability distributions in this setup is their covariance under the entire group of symmetries U. The partition function is invariant against $\mathrm{U}$ transformations, and the set of its arguments, namely the generalized temperatures, can always be reduced to a minimal set whose cardinality is equal to the rank of the compact denominator group $\mathrm{H}\subset \mathrm{U}$. 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

Existing ferroresonance fault identification methods often suffer from high misclassification rates, strong threshold dependency, and insufficient noise resistance. To bridge this gap, we propose a novel ferroresonance fault recognition method based on the Markov transition field (MTF) and three-branch Gaussian clustering (TBGC). Firstly, a symplectic geometric algorithm is employed to denoise the resonance feature signal, extract effective dominant modes, and reshape the series. Secondly, the reshaped feature series is converted into a Pixel matrix image employing the MTF. Subsequently, the gray-level co-occurrence matrix (GLCM) is utilized to extract the two-dimensional texture features of MTF images corresponding to different resonance types and construct corresponding TBGC models. Finally, the overvoltage sequence to be recognized is input into the TBGC model after feature extraction, and accurate discrimination of ferroresonance types is achieved based on cosine similarity. The analysis of fault recording data indicates that this method achieves 100% discrimination accuracy in eight test cases, surpassing the comparative method (maximum accuracy of 62.5%) by 37.5%, thereby validating its effectiveness and accuracy in ferroresonance identification. Full article

Linear vector fields on Lie groups constitute a fundamental class of dynamical systems, as their flows are one-parameter subgroups of automorphisms and their infinitesimal behavior is entirely determined by derivations of the Lie algebra. When a Lie group is endowed with a Hamiltonian-type geometric structure, a natural problem is to determine whether such linear dynamics admit a global variational realization, and how such realizations can be interpreted in terms of reduced models of fluid motion. In the even-dimensional case, where the Lie group carries a symplectic structure, we establish a complete global criterion for the existence of Hamiltonians generating linear symplectic vector fields. The problem reduces to a single global obstruction: the de Rham cohomology class of the 1-form ${\iota }_{\mathcal{X}}\omega$. Thus, every linear symplectic vector field on a simply connected Lie group is globally Hamiltonian, and when the obstruction vanishes, we provide an explicit constructive procedure to recover the Hamiltonian. On the affine group ${\mathrm{Aff}}^{+}\left(1\right)$, this yields a fully explicit, finite-dimensional Hamiltonian model of a 1D ideal fluid with affine symmetries. We then treat odd-dimensional Lie groups, where symplectic geometry is unavailable. Using contact geometry as the canonical replacement, we prove a Hamiltonian lifting theorem ensuring the existence and uniqueness of the associated dynamics. The Reeb vector field appears as a distinguished vertical direction resolving the ambiguities of degenerate Hamiltonian systems. On the Heisenberg group $H_3$, this gives a fully explicit contact Hamiltonian model of an effective non-conservative fluid mode. Finally, we interpret symplectic and contact theories within a unified geometric framework and discuss their relevance to geometric formulations of ideal (symplectic) and effective (contact) fluid equations on Lie groups. 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 discuss the geometry behind the classical Heisenberg model at the level suitable for third- or fourth-year students who did not have the opportunity to take a course on differential geometry. The arguments presented here rely solely on elementary algebraic concepts such as vectors, dual vectors and tensors, as well as Hamiltonian equations and Poisson brackets in their simplest form. We derive Poisson brackets for classical spins, along with the corresponding equations of motion for the classical Heisenberg model, starting from the two-sphere geometry, thereby demonstrating the relevance of standard canonical procedures in the case of the Heisenberg model. 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

It has been reported that quantum correction modifies the topological charges of Anti-de-Sitter Reissner–Nordström (AdS-RN) black holes in Kiselev spacetime, yielding new perspectives on topological classification. This leads us to focus on how quantum corrections and other parameters collectively influence the long-term dynamic evolution of strings. First, we analytically examine whether the strings’ motion violates the Maldacena–Shenker–Stanford (MSS) bound. Then, we employ numerical integration to study the influence of various parameters on string chaotic dynamics. Our results demonstrate that the quantum-correction parameter a, the normalization factor c, and black-hole charge Q significantly influence chaotic behavior and the violation of the MSS bound. In particular, as a increases, the system undergoes an order–chaos–order transition, whereas an increase in c or a decrease in Q drives the system from order to chaos. Full article