Search Results (30)

We develop a unified Koopman–von Neumann (KvN) operator and Weyl–Wigner phase-space framework for inviscid ideal (barotropic) Euler flows. Our approach reformulates the nonlinear fluid dynamics as a linear KvN evolution on an enlarged field phase space, thereby enabling us to apply tools developed for quantum mechanics (Weyl quantization, Moyal ⋆-products, and Wigner functionals) to a classical fluid. We construct the appropriate KvN generator (including the required Jacobian term for unitarity) and derive the evolution equation for the corresponding Wigner functional. This framework clarifies when the classical Liouville (Vlasov) description is exact—namely, in quadratic or linear regimes where the Moyal bracket reduces to the Poisson bracket—and when higher-order quantum-like corrections become significant in fully nonlinear regimes. As an analytic example, we obtain a closed-form Wigner solution for a one-dimensional Burgers flow (pressureless Euler) and verify, term by term, that it reproduces the expected Liouville transport (with distributional contributions at the shock). We also compare the phase-space approach with a kinetic (Vlasov–monokinetic) formulation and outline the extension of the framework to three-dimensional flows using a Clebsch variable representation. Full article

We obtain improved regularity estimates on solutions of partial differential equations by combining the method of Fuchsian Reduction with geometric transformations. Examples include the meron problem and the regularity of the conformal radius. In each case, Reduction needs to be combined with a reinterpretation of the underlying geometry. We argue that the geometric meaning assigned to a problem has an influence, positive or negative, on the range of methods envisioned for its solution, and that the Euler–Poisson–Darboux (EPD) equation cannot be properly understood within a single geometric framework. This explains the central position of EPD-like equations. Full article

This study addresses the challenge of consistently transferring atomistic parameters of the C–C bond into phenomenological material characteristics within the framework of continuum mechanics. Particular attention is given to determining the effective transverse diameter of the covalent C–C bond in carbon nanostructures. The dependence of this diameter on Poisson’s ratio ν is examined, and the influence of the interatomic stiffness constants ${\mathrm{k}}_{\mathrm{r}},{\mathrm{k}}_{\mathsf{\theta }}\mathrm{and}{\mathrm{k}}_{\tau }$ is systematically analyzed. Classical representative-volume models of the C–C bond based on the Euler–Bernoulli beam hypothesis violate thermodynamic stability conditions and lead to nonphysical Poisson’s ratio values exceeding 0.5, due to the neglect of shear deformation effects. To overcome this limitation, an approach based on Timoshenko beam theory is proposed, accounting for both bending and shear deformations. This approach enables estimation of energetically equivalent states between the phenomenological representative volume and the corresponding atomistic C–C bond model. As a result, a sixth-order algebraic equation is derived linking the effective bond diameter, the Poisson’s ratio, and the molecular mechanics force constants. Analysis of this equation reveals a narrow range of effective bond diameters and Poisson’s ratios for which thermodynamic stability conditions are satisfied. Within this range, physically consistent macroscopic material parameters can be directly expressed in terms of atomistic force constants. Full article

In the classical problem of the motion of a rigid body around a fixed point, which is described by the Euler–Poisson equations, we propose a new method for computing cases of integrability: first, we provide algorithms for computing values of parameters ensuring potential integrability, and then we select cases of global integrability. By this method we have obtained all the known cases of global integrability and six new cases of potential integrability for which the absence of their global integrability is proven. Full article

The truncated Euler–Maruyama (EM) method for stochastic differential equations with Poisson jumps (SDEwPJs) has been proposed by Deng et al. in 2019. Although the finite-time $L^r$-convergence theory has been established, the strong convergence theory remains absent. In this paper, the strong convergence refers to the use of an $L^2$ measure and places the supremum over time inside the expectation operation. Our version can be used to justify the method within Monte Carlo simulations that compute the expected payoff of financial products. Noting that the conditions imposed are too strict, this paper presents an existence and uniqueness theorem for SDEwPJs under general conditions and proves the convergence of the truncated EM method for these equations. Finally, two examples are considered to illustrate the application of the truncated EM method in option price calculation. Full article

One of the fundamental problems of spacecraft dynamics related to ensuring its angular orientation in the basic coordinate system is considered. The problem of electrodynamic attitude control for a spacecraft in an elliptical near-Earth Keplerian orbit is studied. A mathematical model describing the attitude dynamics of the spacecraft under the action of the Lorentz torque, the magnetic interaction torque, and the gravitational torque is constructed. The multipole model of the Earth’s magnetic field is used. The possibility of electrodynamic attitude control for the spacecraft’s angular stabilization in the orbital frame is analyzed based on the Euler–Poisson differential equations. The problem of electrodynamic compensation of disturbing torque due to the orbit eccentricity is solved. The control strategy for spacecraft electrodynamic attitude stabilization is presented. Electromagnetic parameters that allow stabilizing the spacecraft’s attitude position in the orbital frame are proposed. The disturbing gravity gradient torque is taken into account. The convergence of the control process is verified by computer modeling. Thus, the possibility and advisability of using the electrodynamic method for the spacecraft attitude control and its angular stabilization in the orbital coordinate system in an elliptical orbit is shown. Full article

The paper considers the a posteriori error estimates for fully discrete approximations of time-dependent Poisson–Nernst–Planck (PNP) equations, which provide tools that allow for optimizing the choice of each time step when working with adaptive meshes. The equations are discretized by the Backward Euler scheme in time and conforming finite elements in space. Overcoming the coupling of time and the space with a full discrete solution and dealing with nonlinearity by taking G-derivatives of the nonlinear system, the computable, robust, effective, and reliable space–time a posteriori error estimation is obtained. The adaptive algorithm constructed based on the estimates realizes the parallel adaptations of time steps and mesh refinements, which are verified by numerical experiments with the time singular point and adaptive mesh refinement with boundary layer effects. Full article

Equations of a heavy rotating body with one fixed point can be deduced starting from a variational problem with holonomic constraints. When applying this formalism to the particular case of a Lagrange top, in the formulation with a diagonal inertia tensor the potential energy has a more complicated form as compared with that assumed in the literature on dynamics of a rigid body. This implies the corresponding improvements in equations of motion. Therefore, we revised this case, presenting several examples of analytical solutions to the improved equations. The case of precession without nutation has a surprisingly rich relationship between the rotation and precession rates, which is discussed in detail. Full article

We start with the cosmic Friedmann equations, where we adopt a novel perspective rooted in a Lagrangian formulation, grounded in Newtonian mechanics and the first law of thermodynamics. Our investigation operates under the assumption that the universe is populated by either a perfect fluid or a scalar field. By elucidating the intricate interplay between the Lagrangian formulation and the cosmic Friedmann equations, we uncover the fundamental principles governing the universe’s dynamics within the framework of these elemental constituents. In our concluding endeavor, we embark on the task of harmonizing the classical equations—namely, the conservation, Euler, and Poisson equations—with the principles of General Relativity. This undertaking seeks to extend these foundational equations to encompass the gravitational effects delineated by General Relativity, thus providing a comprehensive framework for understanding the behavior of matter and spacetime in the cosmic context. Full article

Euler–Poisson equations of a charged symmetrical body in external constant and homogeneous electric and magnetic fields are deduced starting from the variational problem, where the body is considered as a system of charged point particles subject to holonomic constraints. The final equations are written for the center-of-mass coordinate, rotation matrix and angular velocity. A general solution to the equations of motion is obtained for the case of a charged ball. For the case of a symmetrical charged body (solenoid), the task of obtaining the general solution is reduced to the problem of a one-dimensional cubic pseudo-oscillator. In addition, we present a one-parametric family of solutions to the problem in elementary functions. Full article

We single out a class of Lagrangians on a group manifold, for which one can introduce non-canonical coordinates in the phase space, which simplify the construction of the Poisson structure without explicitly calculating the Dirac bracket. In the case of the $SO\left(3\right)$ manifold, the application of this formalism leads to the Poincaré–Chetaev equations. The general solution to these equations is written in terms of an exponential of the Hamiltonian vector field. Full article

The existence of traveling waves in an inhomogeneous medium is a vital problem, the solution of which can help in modeling the wave propagation over long distances. Such waves can be storm waves or tsunami waves in the seas and oceans. The presence of solutions in the form of traveling waves indicates that the wave propagates without reflection and, therefore, can transfer energy over long distances. Traveling waves within the framework of the 1D variable-coefficient wave equation exist only for certain configurations of an inhomogeneous medium, some of which can be found by transforming the original equation to the Euler–Darboux–Poisson equation. The solution of the last equation for certain parameter values is expressed in elementary functions, which are the sum of waves running in opposite directions. The mathematical features of such a transformation are discussed in this paper. Full article

This paper primarily concerns the Whitham modulation equation of the complex modified Korteweg–de Vries (cmKdV) equation with a step-like initial value. By utilizing the Lax pair, we derive the N-genus Whitham equations via the averaging method. The Whitham equation can be integrated using the hodograph transformation. We investigate Krichever’s algebro-geometric scheme to propose the averaging method for the cmKdV integrable hierarchy and obtain the Whitham velocities of the integrable hierarchy and the hodograph transformation. The connection between the equations of the Euler–Poisson–Darboux type linear overdetermined system, which determines the solutions of the hodograph transformation, is constructed through Riemann integration, which demonstrates that the Whitham equation can be solved. Finally, a step-like initial value problem is solved and an exotic wave pattern is discovered. The results of direct numerical simulation agree well with the Whitham theory solution, which shows the validity of the theory. Full article

Dynamic stability is an important mechanical behavior of nanobeams, which has been studied extensively using the Euler–Bernoulli and Timoshenko beam theories, while the Levinson-beam-theory-based dynamic instability analysis of nanobeams has not been investigated yet. Shear deformation is not or is not suitably considered in the Euler–Bernoulli and Timoshenko theories, so it is very important to introduce the Levinson beam theory in the dynamic stability analysis of nanobeams, which correctly models the combined action of bending and shear in nanobeams with smaller length/height ratios. In this work, the equation of the transverse vibration of a Levinson beam embedded in an elastic foundation is firstly formulated based on the displacement field of Levinson beam theory, and the nonlocal theory is further applied to the Levinson nanobeam. Then, the governing equation of the dynamic stability of the Levinson nanobeam is derived using Bolotin’s method to achieve a generalized eigenvalue problem corresponding to the boundaries of regions of dynamic instability. The principal instability region (PIR) is the most important among all regions, so the boundary of the PIR is focused on in this work to investigate the dynamic stability of the Levinson nanobeam. When the width, length/height ratio, density, Young’s modulus, Poisson’s ratio, size scale parameter, and medium stiffness increase by about 1.5 times, the width of the PIR changes by about 19%, −57%, −20%, 65%, 0, −9%, and −11%, respectively. If a smaller critical excitation frequency and narrower width of the PIR correspond to the better performance of dynamic stability, the study shows that the dynamic stability of the Levinson nanobeam embedded in an elastic medium improves under a larger length and density and a smaller width, height, and Young’s modulus, since these factors are related to the natural frequency of the nanobeam which controls the width of the PIR. Additionally, the local model would overestimate the dynamic stability behavior of the Levinson nanobeam. Full article

This article investigates the size dependent on piezoelectrically layered perforated nanobeams embedded in an elastic foundation considering the material Poisson’s ratio and the flexoelectricity effects. The composite beam is composed of a regularly squared cut-out elastic core with two piezoelectric face sheet layers. An analytical geometrical model is adopted to obtain the equivalent geometrical variables of the perforated core. To capture the Poisson’s ratio effect, the three-dimensional continuum mechanics adopted to express the kinematics are kinetics relations in the framework of the Euler–Bernoulli beam theory (EBBT). The nonlocal strain gradient theory is utilized to incorporate the size-dependent electromechanical effects. The Hamilton principle is applied to derive the nonclassical electromechanical dynamic equation of motion with flexoelectricity impact. A closed form solution for resonant frequencies is obtained. Numerical results explored the impacts of geometrical and material characteristics on the nonclassical electromechanical behavior of nanobeams. Obtained results revealed the significant effects of the mechanical, electrical, and elastic foundation parameters on the dynamic behavior of piezoelectric composite nanobeams. The developed procedure and the obtained results are helpful for many industrial purposes and engineering applications, such as micro/nano-electromechanical systems (MEMS) and NEMS. Full article