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The dynamic response of dielectric elastomers is widely used in many functional devices, but current research has neglected the effect of varying dielectric permittivity on their dynamic oscillations and stability. This paper studies the thin-walled dielectric balloon in which the stretch-dependent dielectric permittivity is considered. We obtain the dynamic equation of motion by Hamilton's principle. Based on the principle of no energy dissipation in conservative systems, we establish energy conservation at the maximum stretching position and at the initial moment, then we investigate the stability in the dynamic case. It is found that a stretch-related dielectric permittivity can increase the critical electric field of the balloon and can also change the mode of electric field instability and modulate the critical stretch value. In the dynamic case, the stretch-dependent permittivity increases the critical electric field by 4 % when the balloon is only subjected to electric force; moreover, it increases the critical stretch value by 316.68 % by changing the unstable mode from pull-in instability to snap-through instability. It is hoped that this work will provide new thinking in designing functional devices by using the dynamical response and stability of dielectric elastomers. [ABSTRACT FROM AUTHOR]

The given text discusses different approaches to finding equations of motion for physical systems, such as the Newtonian, Lagrangian, and Hamiltonian approaches. It presents a supplementary tool using the total differential of the mechanical energy function to obtain equations of motion for certain systems. The text also explores the relationship between this approach and Hamilton's equations of motion. Additionally, the text provides concise and factual information about different types of motion and their corresponding equations, making it a useful resource for library patrons conducting research on these topics. [Extracted from the article]

We consider the motion of a “magnetic” soliton in two‐component condensates along a nonuniform and time‐dependent background in the framework of Hamiltonian mechanics. Our approach is based on generalization of Stokes' remark that soliton's velocity is related to its inverse half‐width by the dispersion law for linear waves continued to the region of complex wave numbers. We obtain expressions for the canonical momentum and the Hamiltonian as functions of soliton's velocity and transform the Hamilton equations to a Newton‐like equation. The theory is illustrated by several examples of concrete soliton's dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we propose a discrete Hamilton–Jacobi theory for (discrete) Hamiltonian dynamics defined on a (discrete) contact manifold. To this end, we first provide a novel geometric Hamilton–Jacobi theory for continuous contact Hamiltonian dynamics. Then, rooting on the discrete contact Lagrangian formulation, we obtain the discrete equations for Hamiltonian dynamics by the discrete Legendre transformation. Based on the discrete contact Hamilton equation, we construct a discrete Hamilton–Jacobi equation for contact Hamiltonian dynamics. We show how the discrete Hamilton–Jacobi equation is related to the continuous Hamilton–Jacobi theory presented in this work. Then, we propose geometric foundations of the discrete Hamilton–Jacobi equations on contact manifolds in terms of discrete contact flows. At the end of the paper, we provide a numerical example to test the theory. [ABSTRACT FROM AUTHOR]

In a world made of atoms, computer simulations of molecular systems such as proteins in water play an enormous role in science. Software packages for molecular simulation have been developed for decades. They all discretize Hamilton's equations of motion and treat long-range potentials through cutoffs or discretization of reciprocal space. This introduces severe approximations and artifacts that must be controlled algorithmically. Here, we bring to fruition a paradigm for molecular simulation that relies on modern concepts in statistics to explore the thermodynamic equilibrium with an exact and efficient non-reversible Markov process. It is free of all discretizations, approximations, and cutoffs. We explicitly demonstrate that this approach reaches a break-even point with traditional molecular simulation performed at high precision, but without any of its approximations. We stress the potential of our paradigm for crucial applications in biophysics and other fields, and as a practical approach to molecular simulation. We set out a strategy to reach our goal of rigorous molecular simulation. [ABSTRACT FROM AUTHOR]

The specific objective of this study is to analyze the dynamics of functionally graded carbon nanotubes (FGCNT) reinforced double plates. Connected via an elastic layer, the plates have simply supported boundary conditions. In the current study, three carbon nanotubes functionally graded patterns, varying in the thickness direction are considered, including uniformly distributed, functionally graded O-pattern, and functionally graded X-pattern. Following the development of the coupled equations of motion using the Hamilton principle while considering the influences of the elastic layer, the equations are subsequently solved utilizing a two-spatial-variable modal decomposition method. For verification purposes, the equations developed are compared to simplified configurations provided in the existing studies. The solution methodology is verified through comparing against numerical results of simplified configurations of plates obtained from the development of the finite element method and existing studies. Both verifications have shown very good agreement. Influences of plates' dimensions, carbon nanotubes reinforcement, and the stiffness of elastic layer are analyzed and provided in this study. The transverse-motion natural frequencies of the double plates are also identified, and they follow a decreasing trend as the aspect ratio increases for all the cases. The fundamental lateral-motion and axial-motion natural frequency also follows a similar trend as the aspect ratio increases. The reinforcement effect of carbon nanotubes on the transverse-motion natural frequencies is less obvious for thinner plates. An increase in the elastic layer stiffness increases the second series transverse-motion natural frequencies of the double-plate system. Among the considered functionally graded patterns, the functionally graded X-pattern reinforcement provides the largest increase in the transverse-motion natural frequencies. [ABSTRACT FROM AUTHOR]

A free-electron laser is a laser that has the same optical properties as a conventional laser, such as emitting beams of coherent electromagnetic radiation, which can reach high powers, but uses some very different principles of operation to form the beams. The chaotic motion of electrons causes a considerable decrease in the gain and efficiency of free-electron lasers. In this paper, the Hamiltonian of the dual-wiggler free-electron laser with and without axial guide magnetic field is constructed. Hamilton's equations of motion were derived exactly for both cases. The steady-state solution is also derived and investigated so that the initial conditions of the system are clear. The Poincaré surface-of-section maps were plotted after solving Hamilton's equations of motion numerically, where the chaotic behavior of the system was obvious when the axial field was included while the motion became regular and the Hamiltonian is integrable in the absence of the axial guide magnetic field. Regular orbits are observed clearly for large values of the axial field. [ABSTRACT FROM AUTHOR]

The variational problem, Noether symmetry and conserved quantity, and Lie symmetry and conserved quantity of singular systems are investigated on the basis of Riesz derivatives. First, based on Riesz derivatives, the variational problem of Lagrangian systems is studied, the fractional Lagrange equation is established, and the primary constraint problem of the system is discussed when the Lagrangian is singular. Second, the constrained Hamilton equation is established and the compatibility condition is provided. Third, the Noether symmetry and conserved quantity and the Lie symmetry and conserved quantity of the constrained Hamiltonian system are studied. In the end, an example is provided for illustration. [ABSTRACT FROM AUTHOR]

In this study, the efficiency of the exponential rational function method for the new Hamiltonian amplitude equation has been investigated and the results have been graphically tested and analyzed using Mathematica software techniques. What is important is the high efficiency and accuracy of this method and the variety of answers that provide us with a wide range of answers. [ABSTRACT FROM AUTHOR]

Contemporary quantum computers open up novel possibilities for tackling intricate problems, encompassing quantum system modeling and solving partial differential equations (PDEs). This paper explores the optimization of quantum algorithms aimed at resolving PDEs, presenting a significant challenge within the realm of computational science. The work delves into the application of the Variational Quantum Eigensolver (VQE) for addressing equations such as Poisson’s equation. It employs a Hamiltonian constructed using a modified Feynman-Kitaev formalism for a VQE, which represents a quantum system and encapsulates information pertaining to the classical system. By optimizing the parameters of the quantum circuit that implements this Hamiltonian, it becomes feasible to achieve minimization, which corresponds to the solution of the original classical system. The modification optimizes quantum circuits by minimizing the cost function associated with the VQE. The efficacy of this approach is demonstrated through the illustrative example of solving the Poisson equation. The prospects for its application to the integration of more generalized PDEs are discussed in detail. This study provides an in-depth analysis of the potential advantages of quantum algorithms in the domain of numerical solutions for the Poisson equation and emphasizes the significance of continued research in this direction. By leveraging quantum computing capabilities, the development of more efficient methodologies for solving these equations is possible, which could significantly transform current computational practices. The findings of this work underscore not only the practical advantages but also the transformative potential of quantum computing in addressing complex PDEs. Moreover, the results obtained highlight the critical need for ongoing research to refine these techniques and extend their applicability to a broader class of PDEs, ultimately paving the way for advancements in various scientific and engineering domains. [ABSTRACT FROM AUTHOR]

In light of fractional theory, this paper presents several new effective solitonic formulations for the Langmuir and ion sound wave equations. Prior to this study, no previous research has presented the comparision and obtained the generalized fractional soliton solutions of this kind with power law kernel and Mittag-Leffler kernel. The ion sound and Langmuir wave equations are essential in plasma physics, offering insights into the collective behavior of charged particles in plasmas and enabling diagnostics and control of these complex, ionized gas systems. The two distinct fractional order differential operators are substituted for the traditional order derivative to reshape the examined model. The Atangana-Baleanu non-singular and non-local operator and conformable fractional operator are the fractional-order operators that are used to create the fractional complex system equations for Langmuir waves and ion sound. A constructive approach new auxiliary equation method utilizes to obtain the exact analytical soliton solutions for ion sound and Langmuir wave equation. A wide range of soliton solutions is obtained, including mixed complex solitary shock solutions, singular solutions, mixed shock singular solutions, mixed trigonometric solutions, mixed singular solutions, exact solutions, mixed periodic solutions, and mixed hyperbolic solutions, dark soliton, bright soliton, trigonometric solutions, periodic results, and hyperbolic results. The solitons solution of the ion sound and Langmuir wave equations lies in their ability to maintain wave stability, their role in modeling wave propagation and nonlinear effects, their potential use as diagnostic tools, and their relevance in wave-particle interactions in plasma physics. The solitons provide a valuable framework for understanding the behavior of waves in plasmas and offer insights into the complex dynamics of these charged particle systems. A graphical comparison analysis of a few solutions is also shown here, taking into account appropriate parametric values through the use of the software package. Moreover, the results of this study have important implications for Hamilton's equations and generalized momentum, where solitons are employed in long-range interactions. [ABSTRACT FROM AUTHOR]

Based on the Hamilton canonical equations for ocean surface waves with four-five-six-wave resonance conditions, the determinate dynamical equation of four-five-six-wave resonances for ocean surface gravity waves in water with a finite depth is established, thus leading to the elimination of the nonresonant second-, third-, fourth-, and fifth-order nonlinear terms though a suitable canonical transformation. The four kernels of the equation and the 18 coefficients of the transformation are expressed in explicit form in terms of the expansion coefficients of the gravity wave Hamiltonian in integral-power series in normal variables. The possibilities of the existence of integrals of motion for the wave momentum and the wave action are discussed, particularly the special integrals for the latter. For ocean surface capillary–gravity waves on a fluid with a finite depth, the sixth-order expansion coefficients of the Hamiltonian in integral-power series in normal variables are concretely provided, thus naturally including the classical fifth-order kinetic energy expansion coefficients given by Krasitskii. [ABSTRACT FROM AUTHOR]

The common geometrical (symplectic) structures of classical mechanics, quantum mechanics, and classical thermodynamics are unveiled with three pictures. These cardinal theories, mainly at the non-relativistic approximation, are the cornerstones for studying chemical dynamics and chemical kinetics. Working in extended phase spaces, we show that the physical states of integrable dynamical systems are depicted by Lagrangian submanifolds embedded in phase space. Observable quantities are calculated by properly transforming the extended phase space onto a reduced space, and trajectories are integrated by solving Hamilton's equations of motion. After defining a Riemannian metric, we can also estimate the length between two states. Local constants of motion are investigated by integrating Jacobi fields and solving the variational linear equations. Diagonalizing the symplectic fundamental matrix, eigenvalues equal to one reveal the number of constants of motion. For conservative systems, geometrical quantum mechanics has proved that solving the Schrödinger equation in extended Hilbert space, which incorporates the quantum phase, is equivalent to solving Hamilton's equations in the projective Hilbert space. In classical thermodynamics, we take entropy and energy as canonical variables to construct the extended phase space and to represent the Lagrangian submanifold. Hamilton's and variational equations are written and solved in the same fashion as in classical mechanics. Solvers based on high-order finite differences for numerically solving Hamilton's, variational, and Schrödinger equations are described. Employing the Hénon–Heiles two-dimensional nonlinear model, representative results for time-dependent, quantum, and dissipative macroscopic systems are shown to illustrate concepts and methods. High-order finite-difference algorithms, despite their accuracy in low-dimensional systems, require substantial computer resources when they are applied to systems with many degrees of freedom, such as polyatomic molecules. We discuss recent research progress in employing Hamiltonian neural networks for solving Hamilton's equations. It turns out that Hamiltonian geometry, shared with all physical theories, yields the necessary and sufficient conditions for the mutual assistance of humans and machines in deep-learning processes. [ABSTRACT FROM AUTHOR]

Spin–orbit couplings (SOCs), originating from the relativistic corrections in the Dirac equation, offer nonlinearity in the classical limit and are capable of driving chaotic dynamics. In a nanoscale quantum dot confined by a two-dimensional parabolic potential with SOCs, various quantum scar states emerge quasi-periodically in the eigenstates of the system, when the ratio of confinement energies in the two directions is nearly commensurable. The scars, displaying both quantum interference and classical trajectory features on the electron density, due to relativistic effects, serve as a bridge between the classical and quantum behaviors of the system. When the strengths of Rashba and Dresselhaus SOCs are identical, the chaos in the classical limit is eliminated as the classical Hamilton's equations become linear, leading to the disappearance of all quantum scar states. Importantly, the quantum scars induced by SOCs are robust against small perturbations of system parameters. With precise control achievable through external gating, the quantum scar induced by Rashba SOC is fully controllable and detectable. [ABSTRACT FROM AUTHOR]

We study parameterisation-independent closed planar curve matching as a Bayesian inverse problem. The motion of the curve is modelled via a curve on the diffeomorphism group acting on the ambient space, leading to a large deformation diffeomorphic metric mapping (LDDMM) functional penalising the kinetic energy of the deformation. We solve Hamilton's equations for the curve matching problem using the Wu-Xu element (Wu and Xu (2019) [12]) which provides mesh-independent Lipschitz constants for the forward motion of the curve, and solve the inverse problem for the momentum using Bayesian inversion. Since this element is not affine-equivalent we provide a pullback theory which expedites the implementation and efficiency of the forward map. We adopt ensemble Kalman inversion (EKI) using a negative Sobolev norm mismatch penalty to measure the discrepancy between the target and the ensemble mean shape. We provide several numerical examples to validate the approach. [ABSTRACT FROM AUTHOR]

We develop a method for deriving Hamilton's equations describing the dynamics of solitons when they move along an inhomogeneous and time-varying large-scale background for nonlinear wave equations that are completely integrable in the Ablowitz–Kaup–Newell–Segur (AKNS) scheme. The method is based on the development of old Stokes' ideas that allow analytically continuing the relations for linear waves into the soliton region, and is practically implemented in the example of the defocusing nonlinear Schrödinger equation. A condition is formulated under which the external potential is only to be taken into account when describing the evolution of the background, and that this case, the Newton equation is obtained for the soliton dynamics in an external potential. [ABSTRACT FROM AUTHOR]

This study presents an approach for precisely determining the stretching vibrational frequencies of the P-Cl bond in phosphorus trichloride (PCl3) using a vibrational Hamiltonian framework that maintains the C3v symmetry point group. Our methodology enables the accurate prediction of vibrational frequencies up to the fifth overtone. It identifies related combination bands, marking a significant advancement in vibrational spectroscopy and molecular modelling. By enhancing the accuracy and depth of our understanding of molecular vibrations, this research paves the way for developing more sophisticated computational models, thereby significantly improving the precision of chemical analyses, and contributing to the broader field of chemical physics. [ABSTRACT FROM AUTHOR]

Two systems of mathematical physics are defined by us, which are the first-order differential system (FODS) and the second-order differential system (SODS). Basing on the conventional Legendre transformation, we obtain a new kind of canonical equations of Hamilton (CEH) with some kind of symmetry. We show that the FODS can only be expressed by the new CEH, but do not by the conventional CEH, while the SODS can be done by both the new and the conventional CEHs, on basis of the same conventional Legendre transformation. As an example, we prove that the nonlinear Schrödinger equation can be expressed with the new CEH in a consistent way. Based on the new CEH, the approximate soliton solution of the nonlocal nonlinear Schrödinger equation is obtained, and the soliton stability is analysed analytically as well. Furthermore, because the symmetry of a system is closely connected with certain conservation theorem of the system, the new CEH may be useful in some complicated systems when the symmetry considerations are used. [ABSTRACT FROM AUTHOR]

Purpose: Studying the vibration behavior of functionally graded material (FGM) sandwich structures is crucial for their applications in aerospace and other fields. However, there is little research on the vibration behavior of functionally graded material soft core sandwich plates under different boundary conditions. The purpose of this study is to investigate the free vibration of functionally graded material sandwich plates with soft core layers under different boundary conditions, and to explore the influence of different parameter changes on the natural frequency of functionally graded material soft core sandwich plates under different boundary conditions. Methods: Based on the theory of refine shear deformation, a new free vibration model of FGM soft core sandwich plates under different boundary conditions with only four unknown variables is proposed. There are only four unknown variables for the motion equation and model. Derive the equation of motion through Hamilton's principle. Based on the motion equation, analytical solutions for FGM soft core sandwich plates under various boundary conditions are obtained. The analytical solution of free vibration of Al/Al2O3 FGM soft core sandwich plates under simply supported boundary conditions was compared with the results in the literature to verify the accuracy of the model proposed in this paper. In addition, two different shape functions were used to analyze the free vibration of FGM soft core sandwich plates under different boundary conditions. Results: The dimensionless natural frequencies of FGM soft core sandwich plates under different boundaries will increase with the increase of side-to-thickness ratio and volume fraction index, and decrease with the increase of aspect ratio. Moreover, the dimensionless natural frequencies of symmetric FGM soft core sandwich plates under different boundary conditions are higher than those of asymmetric FGM soft core sandwich plates under different boundary conditions. Conclusion: The changes in different parameters have different effects on the free vibration of three sets of FGM soft core sandwich plates with different material groups under different boundary conditions. However, regardless of the parameter changes, the natural frequencies of these three different material groups of FGM soft core sandwich plates are the highest under FCFC boundary conditions and the lowest under SSSS boundary conditions. [ABSTRACT FROM AUTHOR]

The ideal magnetohydrodynamics (MHD) as well as the ideal fluid dynamics is governed by a Hamilton equation with respect to the Lie–Poisson bracket. The Nambu bracket manifestly represents the Lie–Poisson structure in terms of derivatives of the Casimir invariants. We construct a compact Nambu bracket representation for the 3D ideal MHD equations with the use of three Casimirs for the second Hamiltonians, the total entropy, and the magnetic and cross-helicities, whose coefficients are all constant. The Lie–Poisson bracket induced by this Nambu bracket does not coincide with the original one, but is supplemented by terms with an auxiliary variable. The supplemented Lie–Poisson bracket qualifies the cross-helicity as the Casimir. By appealing to Noether's theorem, we show that the cross-helicity is an integral invariant associated with the particle-relabeling symmetry. Employing a Lagrange label function as the independent variable in the variational framework facilitates implementation of the relabeling transformation. By incorporating the divergence symmetry, other known topological invariants are put on the same ground as Noether's theorem. [ABSTRACT FROM AUTHOR]

The work focuses on the dynamic stability problem of a simply supported three-layer beam subjected to a pulsating axial force. Two analytical models of this beam are developed: one model takes into account the non-linear hypothesis of cross-section deformation, and the other takes into account the standard "broken line" hypothesis. Displacements, strains and stresses for each model are formulated in detail. Based on the Hamilton principle, equations of motion are determined for each of these models. These systems of two differential equations for each model are approximately solved with the consideration of the axial pulsating force, and the fundamental natural frequencies, critical forces and the Mathieu equation are determined. Detailed studies are performed for an exemplary family of beams. The stable and unstable regions are calculated for the three pulsating load cases. The values of fundamental natural frequencies and critical forces of exemplary beams calculated from two models are compared. [ABSTRACT FROM AUTHOR]

In this work, the nonlinear vibrations of functionally graded (FG) porous circular plates under hygro-thermal loading is studied utilizing a numerical approach. Hygroscopic stresses generated due to the nonlinear rise in moisture concentration, even and uneven porosity distributions, and temperature dependency of material properties are all taken into account. Modified Voigt's rule of mixture is applied to obtain the hygro-thermo-mechanical properties of the FG circular plate. All material properties are assumed to be temperature-dependent using the Touloukian formula. In order to obtain the equations of motion, the first-order shear deformation theory, von-Kármán geometrical non-linearity assumption, and hygro-thermal strains are considered concurrently. After deriving the equations of motion using Hamilton's principle, the differential quadrature method and the Newmark-beta time integration scheme are employed in conjunction with an iterative approach to solve the set of nonlinear governing differential equations of motion. The effects of various parameters including temperature distribution, plate's thickness, porosity volume fraction, moisture concentration, and FG index are studied on plate's maximum non-dimensional lateral deflection. [ABSTRACT FROM AUTHOR]

The local scaling symmetry of the Lagrange density is exploited to investigate the myriad electro-mechanical coupling effects observed in the elastic dielectrics. In contrast to most known flexoelectricity theories, this approach has also explicated on the geometric underpinnings of the induced polarization and electric potential. Due to the inhomogeneous scaling of the metric, the invariance of the Lagrange density will be lost. To restore the gauge invariance of the Lagrange density, we introduce the notion of minimal replacement, viz., define a gauge covariant operator in place of the ordinary partial derivative. Minimal replacement introduces gauge compensating one form field. We relate the exact part of the gauge compensating one form field with electric potential. However, the gauge compensating one form field's anti-exact part is correlated with the electric polarization vector. We constructed different components of the gauge invariant energy density using the geometric objects like scale invariant gauge curvature. We appropriately contracted the skew symmetric gauge curvature tensor with the metric to construct the scale invariant energy associated with the gauge field. Finally, we derive the governing coupled equations employing Hamilton's principle. In order to assess how the theory performs, we carry out a few numerical simulations and validation. Model prediction of the pressurized flexoelectric disk with a central hole shows good agreement with the analytical solution. We also investigated flexoelectric cantilever beam subjected to a point load and different earthing conditions. Explorations of this kind of coupling may have notable implications in many industrial and laboratory applications. • Studied electro-mechanical coupling in dielectrics using the scaling symmetry. • Explored geometric underpinnings of induced polarization and electric potential. • Also recovered elastic Lagrangian assuming Euclidean manifold with gauge fixing. • Correspondence with Mindlin's polarization gradient theory for dielectrics. • This kind of coupling may have notable implications in many applications. [ABSTRACT FROM AUTHOR]

The dynamics of the spring-pendulum system with two degrees of freedom were studied. The motion of this system is restricted to be in a vertical plane so that the chosen generalized coordinates are the increased length of the spring and the swing angle of pendulum. Hamiltonian of the system is obtained from the Legendre transformation of Lagrangian. Hamilton's equation yields four differential equations that represent the dynamic of the system. The obtained results were visualized in configuration space and phase space trajectories. It is shown that generally the greater the initial swing angle, the more complex pattern will occur followed by the appearance of chaotic phenomena. [ABSTRACT FROM AUTHOR]

For the particular case of an elastic multibody system (MBS) that can be modeled using one-dimensional finite elements, the main methods offered by analytical mechanics in its classical form for analysis are presented in a unitary description. The aim of the work is to present in a unitary form the main methods offered by classical mechanics for the analysis of solid systems. There is also a review of the literature that uses and highlights these methods, which need to be reconsidered considering the progress of the industry and the complexity of the studied systems. Thus, the kinematics of a finite element is described for the calculation of the main quantities used in the modeling of multibody systems and in analytical mechanics. The main methods used in the research of MBS systems are presented and analyzed. Thus, Lagrange's equations, Gibbs–Appell equations, Maggi's formalism, Kane's equations and Hamilton's equations are studied in turn. This presentation is determined by the advantages that alternatives to Lagrange's equations can offer, which currently represent the method most used by researchers. [ABSTRACT FROM AUTHOR]

The Hamiltonian Monte Carlo algorithm is known to be highly efficient when sampling high‐dimensional model spaces due to Hamilton's equations guiding the sampling process. For weakly non‐linear problems, linearizing the forward problem enhances this efficiency. This study integrates this linearization with geological prior knowledge for optimal results. We test this approach to estimate the source parameters of a 3.4 magnitude induced event that originated in the Groningen gas field in 2019. The source parameters are the event's centroid (three components), its moment tensor (six components), and its origin time. In terms of prior knowledge, we tested two sets of centroid priors. The first set exploits the known fault geometry of the Groningen gas field, whereas the second set is generated by placing initial centroid priors on a uniform horizontal grid at a depth of 3 km (the approximate depth of the gas reservoir). As for the forward problem linearization, we use an approach in which the linearization is run iteratively in tandem with updates of the centroid prior. We demonstrate that, in the absence of a sufficiently accurate initial centroid prior, the linearization of the forward model necessitates multiple initial centroid priors. Eventually, both prior sets yield similar posteriors. Most importantly, however, they agree with the geological knowledge of the area: the posterior peaks for model vectors containing a centroid near a major fault and a moment tensor that corresponds to normal faulting along a plane with a strike almost aligning with that of the major fault. Plain Language Summary: Earthquake source parameters, such as depth, time, and type of faulting, can be estimated using the recordings (or seismograms) of this (induced) earthquake. Being able to do this such that the uncertainty of the estimated parameters is also quantified is particularly valuable. This, however, requires the use of a probabilistic algorithm. A disadvantage of probabilistic algorithms is their computational cost. In this study, we simplify the relationship between the earthquake source parameters and the seismograms to significantly reduce computational costs. Specifically, we demonstrate that the simplified relation between the earthquake source parameters and the earthquake recordings requires the probabilistic algorithm to be provided with a sufficiently accurate (prior) estimate of these very earthquake source parameters. By means of a magnitude 3.4 induced event that originated in the Groningen gas field in 2019, we show that a geologically inspired prior can be helpful to partly overcome this: we use (known) existing faults in the reservoir to kick start the probabilistic algorithm. As such, we recover earthquake source parameters that are in line with subsurface geological information. Key Points: A modified Hamiltonian Monte Carlo (HMC) algorithm is applied to the recordings of a 3.4 magnitude induced event in the Groningen gas fieldMeaningful prior information is required to ensure the modified HMC algorithm samples the correct mode of the posterior probabilityExisting knowledge of the subsurface geology can be incorporated into the prior information of the algorithm [ABSTRACT FROM AUTHOR]

This paper is concerned with practical fixed-time (FT) stabilization problem of discrete-time impulsive switched port-controlled Hamiltonian systems (DISPCH). First, starting with discrete-time port-controlled Hamiltonian systems, a novel controller is presented to achieve practical FT stability of the obtained closed-loop system. Moreover, in order to well handle the abrupt changes at switch moments in practical switched systems, another novel controller is presented in terms of positive-order Lyapunov functions approach and range dwell time method to make discrete-time impulsive switched port-controlled Hamiltonian system practical FT stable. Ultimately, the validity of proposed methods is illustrated by simulations. [ABSTRACT FROM AUTHOR]

Abstract The study of multibody systems with elastic elements involves at the moment the reevaluation of the classical methods of analysis offered by analytical mechanics. Modeling this system with the finite element method requires obtaining the motion equation for an element in the circumstances imposed by a multibody system. The paper aims to present the main analysis methods used by researchers, to make a comparative analysis, and to show the advantages or disadvantages offered by different methods. For the presentation of the main methods (namely Lagrange’s equations, Gibbs–Appell’s equations, Maggi’s formalism, Kane’s equations, and Hamilton’s equations) a unified notation is used. The paper provides a critical evaluation of the studied applications that involved some of these methods, highlighting the reason why it was decided to use them. Also, the paper identifies potential research areas to explore.

Hybrid methods that combine molecular dynamics methods capable of analyzing dynamics with Monte Carlo (MC) methods that can efficiently treat thermodynamically stable states are valuable for understanding complex chemical processes in which an equilibrium state is reached through many elementary processes. The hybrid MC (HMC) method is one such promising method; however, it often fails to sample configurations properly from the canonical multimodal distribution due to the rugged potential energy surfaces. In this paper, we extend the HMC method to overcome this difficulty. The new method, which is termed potential scaling HMC (PS-HMC), makes use of an artificially modulated trajectory to propose a new configuration. The trajectory is generated from Hamilton's equations, but the potential energy surface is scaled to be gradually flattened and then recovered to the original surface, which facilitates barrier-crossing processes. We apply the PS-HMC method to three kinds of molecular processes: the thermal motion of argon particles, butane isomerization, and an atom transfer chemical reaction. These applications demonstrate that the PS-HMC method is capable of correctly constructing the canonical ensemble with a multimodal distribution. The sampling efficiency and accepted trajectories are examined to clarify the features of the PS-HMC method. Despite the potential scaling, many reactive atom transfer trajectories (elementary processes) pass through the vicinity of the minimum energy path. Furthermore, we demonstrate that the method can properly imitate the relaxation process owing to the inherent configurational continuity. By comparing the PS-HMC method with other relevant methods, we can conclude that the new method is a unique approach for studying both the dynamic and thermodynamic aspects of chemical processes. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the relation between the gradient-flow equations and Hamilton's equations in information geometry. By regarding the gradient-flow equations as Huygens' equations in geometric optics, we have related the gradient flows to the geodesic flows induced by the geodesic Hamiltonian in an appropriate Riemannian geometry. The original evolution parameter t in the gradient-flow equations is related to the arc-length parameter in the associated Riemannian manifold by Jacobi–Maupertuis transformation. As a by-product, the relation between the gradient-flow equation and replicator equations is found. [ABSTRACT FROM AUTHOR]

Nondipole terms in the atom–laser interaction arising due to the presence of a magnetic component in an electromagnetic wave and its inhomogeneity lead to the nonseparability of the center-of-mass (CM) and electron variables in the neutral atom and, as a consequence, to its acceleration. We investigate this effect and the accompanying excitation and ionization processes for the hydrogen atom in strong ( 10 12 – 2 × 10 14 W/cm2) linearly polarized short-wavelength (5 eV ≲ ℏ ω ≲ 27 eV) electromagnetic pulses of about 8 fs duration. The study was carried out within the framework of a hybrid quantum-quasiclassical approach in which the coupled time-dependent Schrödinger equation for an electron and the classical Hamilton equations for the CM of an atom were simultaneously integrated. Optimal conditions with respect to the frequency and intensity of the electromagnetic wave for the acceleration of atoms without their noticeable ionization were found in the analyzed region. [ABSTRACT FROM AUTHOR]

In order to focus on the complex effects of porosity distribution and grading pattern on the free vibration response of functionally graded material (FGM) plates, this work employs first-order shear deformation theory to calculate the non-dimensional frequency of different porous FGM plates with simply supported, as well as four types of porosity distribution are given consideration. Assuming that the grading patterns of the plates are different, three functions (pow-law, exponential and sigmoid) are used to obtain the material performance in the specified direction. Derive the equations of motion from Hamilton's principle, which are then precisely solvable using the Navier's method. Analytic solution of vibration frequency is provided, demonstrating the correctness of the suggested methodology by comparison with data from the existing research. A detailed parametric was performed, including porosity, gradient index, width-to-thickness ratio, grading pattern, porosity distribution, and ceramic material. In E-FGM, the frequency of uneven porosity is significantly increased by about 67.3 % compared to even porosity. Most notably in the even porosity, S-FGM frequency values increased by approximately 54 % and 70 % compared to P-FGM and E-FGM, respectively. [ABSTRACT FROM AUTHOR]

In this article, the dynamic characteristics of Co‐curing Damping Sandwich Composites with Different Lay‐up Angles (CDSCDLA) are studied by means of modal analysis and numerical simulation to optimize the structure of co‐curing composites. The vibration equilibrium equation of CDSCDLA is derived according to first order shear deformation theory and Hamilton equation. The dynamic model of the structure is established by finite element simulation to verify the accuracy of the theoretical calculation. The influence of different lay‐up angle and the damping film thickness on the free vibration of CDSCDLA is explored under the condition of constant skin thickness. The theoretical derivation and finite element simulation results reveals that the overall dynamic characteristic of damping sandwich composite with a lay‐up angle of 45° is optimal when the damping film thickness is constant. In particular, when the thickness of the damping film is 0.1 mm, the first‐order mode damping and mode frequency of composite plate with a lay‐up angle of 45° reach the maximum value of 0.1495 and 747.9 Hz, respectively. The first order damping loss factor of the CDSCDLA with different layup angles is maximum when the damping film thickness is 0.3 mm. However, blindly increasing the damping layer thickness can not always improve the damping performance of the composite structure, but will greatly reduce the stiffness of the composite. The work provides a reference for dynamic performance optimization and structural design of sandwich plate with high strength and large damping loss factor. Highlights: Free vibration of sandwich composites with lay‐up angles is investigated.The dynamic analytical model of the structure is established.The validity of the derived theory is verified by numerical simulation. [ABSTRACT FROM AUTHOR]

The canonical transformation and Poisson theory for the second-order generalized mechanical systems based on non-standard power-law Lagrangians are studied. First, the Euler–Lagrange equations and the Hamilton canonical equations for the second-order generalized mechanics with the power-law Lagrangians are established. Second, the canonical transformation theory of the systems is studied by establishing the relationship between old and new variables. Four basic forms of canonical transformation are given, and the transformation formulas in each case are derived. Third, the algebraic structure of the dynamical equations of the systems is studied, and the corresponding Poisson theory is established. Finally, the corresponding examples are presented to illustrate the application of the results we obtained. [ABSTRACT FROM AUTHOR]

The chaotic motion behavior of the rectangular conductive thin plate that is simply supported on four sides by airflow and mechanical external excitation in a magnetic field is studied. According to Kirchhoff’s thin plate theory, considering geometric nonlinearity and using the principle of virtual work, the nonlinear motion partial differential equation of the rectangular conductive thin plate is deduced. Using the separate variable method and Galerkin’s method, the system motion partial differential equation is converted into the general equation of the Duffing equation; the Hamilton system is introduced, and the Melnikov function is used to analyze the Hamilton system, and obtain the critical surface for the existence of chaos. The bifurcation diagram, phase portrait, time history response and Poincaré map of the vibration system are obtained by numerical simulation, and the correctness is demonstrated. The results show that when the ratio of external excitation amplitude to damping coefficient is higher than the critical surface, the system will enter chaotic state. The chaotic motion of the rectangular conductive thin plate is affected by different magnetic field distributions and airflow. [ABSTRACT FROM AUTHOR]

The study of multibody systems with elastic elements involves at the moment the reevaluation of the classical methods of analysis offered by analytical mechanics. Modeling this system with the finite element method requires obtaining the motion equation for an element in the circumstances imposed by a multibody system. The paper aims to present the main analysis methods used by researchers, to make a comparative analysis, and to show the advantages or disadvantages offered by different methods. For the presentation of the main methods (namely Lagrange's equations, Gibbs–Appell's equations, Maggi's formalism, Kane's equations, and Hamilton's equations) a unified notation is used. The paper provides a critical evaluation of the studied applications that involved some of these methods, highlighting the reason why it was decided to use them. Also, the paper identifies potential research areas to explore. [ABSTRACT FROM AUTHOR]

With the special physical characteristics, graphene is considered as an excellent candidate for various engineering applications. Due to the electromechanical coupling properties and significant variations in material characteristics at the interfaces of adjacent layers, existing higher-order models reported in the literature may lack the necessary capability to achieve precise prediction of natural frequencies for piezoelectric graphene-reinforced composite plates. This paper will develop advanced plate theory for the vibration analysis of graphene-reinforced composite plates with a macro fiber composite piezoelectric layer. The number of displacement variables in the developed theory is independent of the layer number. In contrast to earlier higher-order theories, the proposed plate theory incorporates a modified interlaminar shear stress field that accounts for the electromechanical properties. Furthermore, the modified transverse shear stress field can be adsorbed in the equations of motion by means of Hamilton's principle, which can effectively improve the ability to predict natural frequencies of piezoelectric laminated plates. Using the exact solutions and the results obtained from other theories, the performance of the developed theory is evaluated. Compared with the existing higher-order models, the proposed theory is more accurate in predicting natural frequencies. Additionally, a parametric study is conducted for the influences of several significant parameters of graphene and the piezoelectric plate on the vibration responses of the smart composite plates with graphene reinforcements. [ABSTRACT FROM AUTHOR]

By applying the Keeton and Petters technique, we find the deflection angle as a series expansion with a specific factor of mass. We investigate the shadow structure of a regular black hole with cosmic string with a non-magnetized, pressure-free plasma around it. In order to separate the Hamilton equation and locate the photon areas, certain plasma distributions are taken into consideration. Moreover, by utilizing a new ray-tracing technique, we analyze the photon's path around a regular black hole with cosmic string in the presence of plasma whose density depends on the radius coordinate. We also analyze the plasma effects on the shadow of black hole. We obtain an analytical formula that describes the boundary curve of the shadow for such a black hole in an expanding universe for an observer at any finite location outside the horizon. In the presence and absence of plasma, we detect deviations that are further investigated by analyzing the geometry of shadow angle at a particular value of plasma frequency. [ABSTRACT FROM AUTHOR]

The problem of diffraction of slow neutrons with a de Broglie wavelength λ ~10-10 m in a crystal with a deformed lattice is considered. Based on the Hamilton equations describing the dynamic diffraction of slow neutrons, the features of the formation of a dynamic diffraction field in a lattice in various regions of the deformation field characteristics, including weak fields, slowly changing fields and rapidly changing fields are discussed. Using a particular example of a deformation field with a displacement field quadratic in coordinates, analytical solutions are obtained for the wave functions of transmitted and reflected lattice waves. [ABSTRACT FROM AUTHOR]

The paper develops the methods for computer simulation of the dynamics of absolutely rigid body systems with a tree structure. The equations of motion are presented in compact matrix form. The case of holonomic constraints is also considered. Since the independent parameters uniquely determine the positions and velocities of the bodies of the system in space, the generalized coordinates and variables with the dimension of impulses are chosen. A feature of the system of equations is that it is resolved with respect to the derivatives of the generalized momenta and does not contain constraint reactions. The derivation of the proposed form of the equations of motion from the Hamilton principle using the matrix-geometric approach is given. Recursive formulas for determining all kinematic and dynamic variables included in the equations are obtained. As an example of a mechanical system with six degrees of freedom, all stages of preparing primary information and compiling equations of motion in the proposed form are demonstrated. Three algorithms are presented that allow resolving the obtained extended system of equations with respect to the generalized velocities and momenta without direct formation of the Hamilton equations. The classification of the equations of motion of rigid body systems is carried out from their structure point of view. The place of the deduced equations in the general classification is also demonstrated. A comparative analysis of the computational complexity of the considered methods for various classes of mechanical systems is carried out. Some diagrams are constructed that allow choosing the most effective modeling method depending on the characteristics of the mechanical system: the number of bodies, the number of degrees of freedom and the structure of the system. [ABSTRACT FROM AUTHOR]

In this article, dynamic analysis of viscoelastic beam, which is made of porous functionally graded materials (FGM), is investigated by using the generalised differential quadrature method (GDQM). The material properties for FG beam are supposed to be graded in the thickness direction by two types of porosities and estimated using a modified power-law form. The surrounding visco-Pasternak foundation is made of viscous layer symbolised by dashpots, parallel springs and a shear layer. The governing equations achieved by Hamilton's principle are solved implementing GDQM. Afterwards, the effects of types of porosity distribution and its pattern, foundation type, power-law index, slenderness ratio, structural damping coefficient and boundary conditions on the damping frequency of the present viscoelastic FG beams are explored. [ABSTRACT FROM AUTHOR]

This paper develops a new approach to small time local attainability of smooth manifolds of any dimension, possibly with boundary and to prove Hölder continuity of the minimum time function. We give explicit pointwise conditions of any order by using higher order hamiltonians which combine derivatives of the controlled vector field and the functions that locally define the target. For the controllability of a point our sufficient conditions extend some classically known results for symmetric or control affine systems, using the Lie algebra instead, but for targets of higher dimension our approach and results are new. We find our sufficient higher order conditions explicit and easy to compute for targets with curvature and general control systems. Some cases of nonsmooth targets are also included. [ABSTRACT FROM AUTHOR]

The symplectic method for the elastic problem of decagonal symmetric 2D quasicrystal plates with 2 opposite edges simply supported, was discussed. The basic equations of the elastic theory for decagonal symmetric 2D quasicrystals were transformed into the Hamilton dual equations. With the method of separation of variables, the symplectic eigenvalues of the corresponding Hamilton operator matrix and the symplectic eigenfunction system were obtained. The completeness of the symplectic eigenfunction system of the Hamilton operator matrix in the sense of the Cauchy principal value was proved. Based on the symplectic eigenfunction expansion of the Hamilton system, the analytical solution to the bending problem of the decagonal symmetric 2D quasicrystal plate was given. [ABSTRACT FROM AUTHOR]