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Modulation of the refractive index in a medium by external stimuli enables fast and reversible control of light propagation. This technology for controlling light has led to new discoveries in a wide range of research fields from physics to life sciences and has played a major role in the development of photonics devices. In this article, we focus on ultrasound as an external stimulus and have devised a method to control the refractive index of a medium using ultrasound. Our research group has previously discovered that a giant refractive-index gradient (Δn on the order of 10−2) was induced when water was irradiated with high-frequency (100 MHz range), high-intensity (on the order of MPa) ultrasound. Here, we report ray-tracing simulations in a medium with a refractive-index gradient induced by ultrasonic radiation. A numerical model of the refractive-index gradient was developed based on the experimental data, and ray-tracing simulations were performed using the Euler–Lagrange equation. The ray-tracing simulation results were close numerically to the profiles of the laser beam observed in the experiment when the laser beam was incident on the refractive-index-gradient medium. [ABSTRACT FROM AUTHOR]

We present the locally conformal generalization of the Euler–Lagrange equations. We determine the dual space of the LCS Hamiltonian vector fields. Within this dual space, we formulate the Lie–Poisson equation that governs the kinetic motion of Hamiltonian systems in the context of local conformality. By expressing the Lie–Poisson dynamics in terms of density functions, we derive locally conformal Vlasov dynamics. In addition, we outline a geometric pathway that connects LCS Hamiltonian particle motion to locally conformal kinetic motion. [ABSTRACT FROM AUTHOR]

This paper aims at studying synchronization, stability and motion characteristics of a new generalized dynamical model with two rigid bodies (RBs), driven by multiple eccentric rotors (ERs) uniformly distributed on the circumference of two different RBs. Using Lagrange's equation, the motion differential equations of the system are deduced. The generalized theoretical analytical expressions of the criteria for implementing synchronization and stable operation of multiple ERs are obtained. Based on the above theoretical results, and taking the present dynamical model with six ERs for example, numerical qualitative discussions are given to deeply reveal the stable states and motion characteristics of the system in different resonant regions. The simulations are further carried out quantitatively, which verifies the validity of the above theoretical and numerical qualitative results. It is shown that, in the sub-resonant state with respect to the natural frequency of the main vibrating system, the stable zero phase difference between arbitrary two ERs on each RB can be achieved, in this case, the relative circular motion of two RBs with anti-phase is realized. This paper can lay a foundation for designing some new types of large scale high-frequency vibrating ball mills with high supply power. [ABSTRACT FROM AUTHOR]

Considering the displacement-dependent characteristics of the clamp, a general nonlinear semi-analytical modeling method for the spatial pipeline with arbitrary shapes supported by multiple clamps is proposed to study the vibration characteristics of the pipeline-clamp system. The coupling of the complex spatial pipeline is realized by introducing the connecting springs at the connection region of the straight pipe and curved pipe. The equation of motion for spatial pipeline structure is established based on the Lagrange energy equation and Timoshenko beam theory. High-order polynomials are used to characterize the stiffness and damping characteristics of the clamp, and the simple straight pipe is used as the basic model to reverse the clamp parameters. The influence of clamps on pipeline vibration is skillfully introduced into the nonlinear dynamic model of the spatial pipeline. According to the Newton-Raphson method, the nonlinear equation of motion is solved. Taking the spatial pipeline supported by double clamps as an example, the related experiment tests and ANSYS simulation are carried out, and the effectiveness of the established model is verified by comparing the results. Finally, the nonlinear vibration mechanism of the pipeline-clamp system is analyzed. The modeling method proposed in this paper can provide positive guidance for modeling and vibration analysis of the aero-engine pipeline with arbitrary shapes. [ABSTRACT FROM AUTHOR]

This paper is concerned with the tensor sparse principal component analysis (TSPCA) by obtaining principal components which are linear combinations of a small subset of the original features for tensorial data. The core mathematical model can be formulated as a nonsmooth nonconvex optimization problem with a polynomial objective function, and with the sparsity constraint and the unit Euclidean spherical constraint. By employing the tools in tensor analysis, along with the variational properties for the involved $ \ell _{0} $ ℓ 0 -norm, the optimality condition of TSPCA is analysed in terms of stationary points. To well resolve the problem, we reformulate the stationary conditions into the Lagrange stationary equation system via the property of the projection operator onto the sparsity constraint set. With special emphasis on the Jacobian nonsingularity of the corresponding nonlinear system, we propose the Lagrange–Newton algorithm for pursuing the stationary point, which serves as a promising approximation of the optimal solution to TSPCA. The locally quadratic convergence rate is also established under mild conditions. Numerical experiments illustrate the effectiveness of our proposed TSPCA approach in terms of the solution accuracy as well as the computation time. [ABSTRACT FROM AUTHOR]

This paper proposes a Fourier-Legendre spectral method to find the minimizers of a variational problem, called \sigma _{2,p}-energy, in polar coordinates. Let {\mathbb {X}}\subset \mathbb {R}^n be a bounded Lipschitz domain and consider the energy functional (1.1) whose integrand is defined by {\mathbf {W}}(\nabla u(x))≔(\sigma _2(u))^{\frac {p}{2}}+\Phi (\det \nabla u) over an appropriate space of admissible maps, \mathcal {A}_p({\mathbb {X}}). Using Fourier and Legendre interpolation errors, we obtain an error estimate for the energy functional and prove a convergence theorem for the proposed method. Furthermore, we apply the gradient descent method to solve a nonlinear algebraic system which is obtained by discretizing the Euler-Lagrange equations. The numerical experiments are performed to demonstrate the accuracy and effectiveness of our method. [ABSTRACT FROM AUTHOR]

This paper discusses wave propagation in unbounded particle-based materials described by a second-gradient continuum model, recently introduced by the authors, to provide an identification technique. The term particle-based materials denotes materials modeled as assemblies of particles, disregarding typical granular material properties such as contact topology, granulometry, grain sizes, and shapes. This work introduces a center-symmetric second-gradient continuum resulting from pairwise interactions. The corresponding Euler-Lagrange equations (equilibrium equations) are derived using the least action principle. This approach unveils non-classical interactions within subdomains. A novel, symmetric, and positive-definite acoustic tensor is constructed, allowing for an exploration of wave propagation through perturbation techniques. The properties of this acoustic tensor enable the extension of an identification procedure from Cauchy (classical) elasticity to the proposed second-gradient continuum model. Potential applications concern polymers, composite materials, and liquid crystals. [ABSTRACT FROM AUTHOR]

This paper presents a novel shear deformation theory for analyzing porous microbeams' bending, buckling, and free vibration resting on a foundation. The proposed shear function incorporating three kinetic variables satisfies zero-traction boundary conditions on the top and bottom surfaces of the beams and does not require a shear correction factor. The modified couple stress theory accounts for the size-dependent effects, and the governing equations are derived from Lagrange's equation using the proposed shear function. Legendre–Ritz functions are developed to analyze the porous microbeams' buckling, free vibration, and bending behaviors. The effects of material length scale parameter, porosity, span-to-height ratio, boundary condition, and foundation parameter on the mechanical responses of beams are investigated. Numerical results demonstrate the accuracy and efficiency of the proposed theory and can serve as benchmarks for future analysis of porous microbeams on elastic foundations. [ABSTRACT FROM AUTHOR]

In this paper, a three-dimensional dynamic model of a dual-cable winding hoisting system is presented to simulate and analyze its coupled vibrations. The equations of motion of this system are derived based on a substructure method and Lagrange's equations of the first kind, in which the longitudinal–torsional coupled mechanical characteristics of the hoisting cables are considered. Longitudinal and transverse natural frequencies of the system are obtained and studied, and its dynamic responses due to some assumed displacement excitations are calculated. Numerical results agree well with those from ADAMS simulation, and the results have shown that longitudinal and transverse resonances are inevitable, but the transverse resonance has little effect on the conveyance; Slow longitudinal excitations should be avoided, thus avoiding the first-order longitudinal resonance since the system longitudinal vibration is dominated by its first-order mode; the torque in the hoisting cable is mainly induced by its tension under longitudinal–torsional coupled mechanical characteristics of the cable, and a torque release device is suggested to protect the hoisting cable from torsion failure; Torsional and transverse vibrations of the conveyance are been well restricted by the guiding cables even when the guiding cable pre-tension is zero. The results of this paper can be used to improve the parameter design of the ultra-deep shaft hoisting systems. [ABSTRACT FROM AUTHOR]

This work analyzes the dynamic responses of plates with an elastic foundation under a concentrated moving mass. The plate is made of fluid-saturated functionally graded porous material (FFGPM). The material properties of the FFGPM plate are assumed to vary smoothly across the thickness, with symmetric, asymmetric, and uniform patterns for porosity distribution. The governing equations of the FFGPM plate are developed using Reddy's third-order shear deformation plate theory (Reddy's TSDT), Biot's poroelasticity theory, pb2-Ritz formulation, and the Lagrange equation. In the numerical results, the Newmark scheme is used to obtain the deflection response of the FFGPM plate with the traveling mass. Two approaches are considered and discussed, including moving load and moving mass. Moreover, the influences of the porosity coefficient, distribution patterns, different boundary conditions, elastic foundations, geometry parameters, Skempton coefficient, moving mass velocity, and the mass of moving mass on the dynamic characteristics of the FFGPM plate are studied. [ABSTRACT FROM AUTHOR]

Particle breakage in sand significantly influences the stress and deformation predictions of the spherical cavity expansion theories. However, existing theories often overlook or lack a quantitative description of particle breakage, leading to significant errors in the actual results. Based on the Simple Critical State Sand (SIMSAND) model, considering the particle breakage effect, the traditional Euler's problem of the drainage spherical cavity expansion is converted into a set of first-order ordinary differential equations described by Lagrangian. Fontainebleau sand is taken as an example, and the influence of the initial stress of the sand on the stress state and void ratio is studied. An analysis of the relative breakage surrounding the cavity reveals that the primary impact zone extends to a distance thrice the expansion radius. An axisymmetric model of the cone penetration test is established, the analytical and numerical solutions of expansion stress and cone tip resistance are analysed and the correctness of the spherical cavity expansion theory is verified. Furthermore, the cone tip resistance results, derived from the theory of spherical cavity expansion, are consistent with the results of numerical calculations. Notably, when the particle breakage effect is not considered, the calculated results of cone tip resistance significantly increase. [ABSTRACT FROM AUTHOR]

We develop an action principle for producing a single-fluid two-constituent system with dissipation in general relativity. The two constituents in the model are particles and entropy. The particle flux creation rate is taken to be zero, while the entropy creation rate is non-zero. Building on previous work, it is demonstrated that a new term (the proper time derivative of the matter space "metric") is required in the Lagrangian in order to produce terms typically associated with bulk and shear viscosity. Equations of motion, entropy creation rate, and energy–momentum–stress tensor are derived. Using an Onsager approach of identifying thermodynamic "forces" and "fluxes", a model is produced which delivers the same entropy creation rate as the standard, relativistic Navier–Stokes equations. This result is then contrasted with a model generated in the spirit of the action principle, which takes as its starting point a specific Lagrangian and then produces the equations of motion, entropy creation rate, and energy–momentum–stress tensor. Unlike the equations derived from Onsager reasoning, where the analogs of the bulk and shear viscosity coefficients are prescribed "externally", we find that the forms of the coefficients in the second example are a direct result of the specified Lagrangian. Furthermore, the coefficients are shown to satisfy evolution equations along the fluid worldline, also a product of the specific Lagrangian. [ABSTRACT FROM AUTHOR]

In this paper, we use the sub-ODE method to analyze soliton solutions for the renowned nonlinear Klein-Gordon model (NLKGM). This method provides a variety of soliton solutions, including three positive solitons, three Jacobian elliptic function solutions, bright solitons, dark solitons, periodic solitons, rational solitons and hyperbolic function solutions. Applications for these solitons can be found in optical communication, fiber optic sensors, plasma physics, Bose-Einstein condensation and other areas. We also study some numerical solutions by using forward, backward, and central difference techniques. Moreover, we discuss variational integrators (VIs) using the projection technique for NLKGM. We develop a numerical solution for NLKGM using the discrete Euler lagrange equation, the Lagrangian and the Euler lagrange equation. At the end, in various dimensions, covering 3D, 2D, and contour, we will also plot several graphs for the obtained NLKGM solutions. A contour plot is a type of graphic representation that displays a three-dimensional surface on a two-dimensional plane by using contour lines. Each contour line in the plotted function represents one of the function's constant values, mapping the function's value across the plane. This model has been studied across multiple soliton solutions using various methods in the open literature, but this model for VIs and finite deference scheme (FDS) is the first time it has been studied. Within the various numerical techniques accessible for solving Hamiltonian systems, variational integrators distinguish themselves because of their symplectic quality. Here are some of the symplectic properties: symplectic orthogonality, energy conservation, area preservation, and structure preservation. [ABSTRACT FROM AUTHOR]

In this paper, Lie symmetries and Noether symmetries along with the corresponding conservation laws are derived for weakly nonlinear dispersive magnetohydrodynamic wave equations, also known as the triple degenerate derivative nonlinear Schrödinger equations. The main goal of this study is to obtain Noether symmetries of the second-order Lagrangian density for these equations using the Noether symmetry approach with a gauge term. For this Lagrangian density, we compute the conserved densities and fluxes corresponding to the Noether symmetries with a gauge term, which differ from the conserved densities obtained using Lie symmetries in Webb et al. (J. Plasma Phys. 1995, 54, 201–244; J. Phys. A Math. Gen. 1996, 29, 5209–5240). Furthermore, we find some new Lie symmetries of the dispersive triple degenerate derivative nonlinear Schrödinger equations for non-vanishing integration functions K i (t) ( i = 1 , 2 , 3 ). [ABSTRACT FROM AUTHOR]

The research is aimed at creating a methodology for increasing the positioning accuracy of an industrial robot and minimizing the vibration of the robot gripper by applying machine learning based on the developed mathematical model for estimating the positioning error. Two components of positioning accuracy are considered: geometric and kinematic errors and elastic static deformations. The dynamic error in the partial system of motion of the robot manipulator links is analyzed. The equation of partial motions is obtained from Lagrange's differential equation of motion of the II kind. The system of differential equations for the positioning error was solved analytically by Euler's method. An example of modeling the position and orientation error of the gripper due to temperature deformations of the third link for the manipulator scheme is given. An example of the modeling of static deformations and errors of the manipulator with elastic pliability of the robot links is given. An example of dynamic error modeling in a partial system of motion of the robot links is given. The proposed method of modeling robot gripper positioning errors makes it possible to increase the positioning accuracy of the industrial robot and minimize the vibration of the gripper. Having a mathematical model of positioning errors, it is possible to compensate for the positioning error by changing the speed of movement of the gripper reference point before determining the direct kinematic task. [ABSTRACT FROM AUTHOR]

This work deals with f(T,TG) gravity in the background of homogeneous and isotropic flat Friedmann–Lemaître–Robertson–Walker (FLRW) space-time model. The main aim of this work is to examine whether the model supports the observational data or not. By using the Noether symmetry analysis, not only the symmetry vector is obtained, but also the coupling function f(T,TG) in the Lagrangian has been determined. Also, the symmetry analysis identifies a transformation from (a,T,TG) to (u,v,w) in such a way that one of the new variables become cyclic along the direction symmetry vector and as a result the field equations become solvable. Then the solution is analyzed from the observational point of view. Finally, quantum cosmology has been formulated for the present model and the wave function of the universe has been evaluated by solving the Wheeler–DeWitt equation with some assumptions. [ABSTRACT FROM AUTHOR]

A four-dimensional differential Euler-Lagrange equation for continuously distributed materials is derived based on the principle of least action, and instead of Lagrangian, this equation contains the Lagrangian density. This makes it possible to determine the density of generalized fourmomentum in covariant form as derivative of the Lagrangian density with respect to four-velocity of typical particles of a system taken with opposite sign, and then calculate the generalized fourmomentum itself. It is shown that the generalized four-momentum of all typical particles of a system is an integral four-vector and therefore should be considered as a special type of fourvectors. The presented expression for generalized four-momentum exactly corresponds to the Legendre transformation connecting the Lagrangian and Hamiltonian. The obtained formulas are used to calculate generalized four-momentum of stationary and moving relativistic uniform systems for the Lagrangian with particles and vector fields, including electromagnetic and gravitational fields, acceleration field and pressure field. It turns out that the generalized fourmomentum of a moving system depends on the total mass of particles, on the Lorentz factor and on the velocity of the system's center of momentum. Besides, an additional contribution is made by the scalar potentials of the acceleration field and the pressure field at the center of system. The direction of the generalized four-momentum coincides with the direction of four-velocity of the system under consideration, while the generalized four-momentum is part of the relativistic fourmomentum of the system. [ABSTRACT FROM AUTHOR]

We study minimisation problems in L ∞ for general quasiconvex first order functionals, where the class of admissible mappings is constrained by the sublevel sets of another supremal functional and by the zero set of a nonlinear operator. Examples of admissible operators include those expressing pointwise, unilateral, integral isoperimetric, elliptic quasilinear differential, Jacobian and null Lagrangian constraints. Via the method of L p approximations as p → ∞ , we illustrate the existence of a special L ∞ minimiser which solves a divergence PDE system involving certain auxiliary measures as coefficients. This system can be seen as a divergence form counterpart of the Aronsson PDE system which is associated with the constrained L ∞ variational problem. [ABSTRACT FROM AUTHOR]

The homogeneous causal action principle on a compact domain of momentum space is introduced. The connection to causal fermion systems is worked out. Existence and compactness results are reviewed. The Euler–Lagrange equations are derived and analyzed under suitable regularity assumptions. [ABSTRACT FROM AUTHOR]

Both horizontal and vertical ground motion can cause damage to vibration-sensitive equipment. However, most mature seismic isolation devices only isolate horizontal ground acceleration; few devices isolate both vertical and horizontal ground acceleration. In this paper, a novel inertia-type bidirectional isolation system (IBIS) is proposed that comprises independent vertical and horizontal seismic isolators. The IBIS's vertical isolator comprises leverage apparatus with a counterweight, which provides a vertical lifting force in the static state to balance the self-weight of the isolated object, and an additional inertial force in the dynamic state, resulting in higher static but lower dynamic effective vertical stiffness. The horizontal isolator is a conventional sliding-type system comprising a sliding platform and pair of springs. The IBIS's equations of motion are derived by using Lagrange's equation, and the theoretical model is verified through shaking-table test results. Numerical simulation and experimental results show that the IBIS has an anti-resonance property in the vertical direction, leading to higher isolation performance under both near-fault and far-field ground acceleration. The conventional sliding-type horizontal isolator component performs well under far-field ground acceleration but may perform unsatisfactorily under near-fault ground acceleration. [ABSTRACT FROM AUTHOR]

This article aims to study the active control of free and forced vibration of a rotating functionally graded (FG) cylindrical shell using FG piezoelectric patches. Accordingly, this study uses sensors and actuators embedded in the shell's inner and outer surfaces. Considering classical theory and Love relations, ordinary differential equations of the system are extracted through the energy method and Lagrange equation. The validation part of the study involves comparing the results of this study with previous studies of rotating and nonrotating cylindrical shells. In addition, the convergence of vibration responses of the rotating cylindrical shell is investigated. Using the LQR method and considering the appropriate value for the weighting matrices, the system's vibration is attenuated, and the required voltage of the actuators is also obtained. Lastly, the effects of parameters such as position and performance index of piezoelectric on the closed-loop response are studied. The results show that changing piezoelectric properties in the form of a power law lead to a quicker decrease in vibration amplitude. [ABSTRACT FROM AUTHOR]

The dynamic characteristics of gear systems are affected by the sliding friction and time-varying pressure angle. In the previous studies, the sliding friction is usually neglected for simplification, while the pressure angle is always considered as a constant. Therefore, the coupling effect between the sliding friction and pressure angle is not considered. In view of this, a new six degrees-of-freedom model for spur gear systems is proposed. The sliding friction and time-varying pressure angle are emphasized in this model. The differential equations of motion are derived using Lagrange's equation. The Runge–Kutta numerical integration algorithm is applied to gain dynamic response. The proposed model is somewhat more realistic compared to previous models. Furthermore, parametric studies are carried out to reveal the effects of friction coefficient and equivalent shaft-bearing stiffness (directly related to translational motion and time-varying pressure angle). The coupling effect between the sliding friction and pressure angle is verified not only by the mathematical model but also by the dynamic response. The results reveal that both friction coefficient and equivalent shaft-bearing stiffness affect the contact ratio and pressure angle and lead to distinctive dynamic responses. This research can provide a foundation for further study and be used as a reference for gear system design and vibration prediction. [ABSTRACT FROM AUTHOR]

To effectively isolate low-frequency vibrations, we present a rigid–flexible coupling quasi-zero-stiffness (QZS) vibration isolator with high-static-low-dynamic stiffness (HSLDS) characteristics. Specifically, the QZS isolator is realized by the development of a compliant constant-force mechanism, formed by parallelly combining a diamond-shape mechanism and a nonlinear bi-stable beam in parallel. To evaluate performance of the QZS isolator, we derived an analytical force–displacement model and dynamic model based on pseudo-rigid body method and Lagrange's equations. Then, finite element analysis was performed in Workbench to verify theoretical analysis and identify the optimal design parameters. Furthermore, the dynamic responses of the QZS isolator are established with the harmonic balance method. Finally, the relationships among displacement transmissibility and factors including damping, BSB, payload mass, and material property are discussed. The results have shown that our QZS isolator design can effectively isolate vibrations in low frequency. [ABSTRACT FROM AUTHOR]

Variational integrators play a pivotal role in the simulation and control of constrained mechanical systems. Current Lagrange multiplier-free variational integrators for such systems are concise but inevitably face issues with parameterization singularities, hindering global integration. To tackle this problem, this paper proposes a novel method for constructing variational integrators on manifolds without introducing Lagrange multipliers, offering the benefit of avoiding singularities. Our approach unfolds in three key steps: (1) the local parameterization of configuration space; (2) the formulation of forced discrete Euler-Lagrange equations on manifolds; and (3) the construction and implementation of high-order variational integrators. Numerical tests are conducted for both conservative and forced mechanical systems, demonstrating the excellent global energy behavior of the proposed variational integrators. [ABSTRACT FROM AUTHOR]

Scissor-like isolation platform (SIP) with magnetorheological damper (MRD) has been commonly studied and applied successfully in vehicle vibration isolation. This paper concerns passive nonlinear magnetorheological (MR) characteristics of the SIP via geometric nonlinearity induced by MRD's layout ways. A dynamic parametric model of the SIP with six assembly types is derived based on Lagrange equation. Then, the parameter analysis is performed to estimate MR damping function in SIP. The analytical steady-state response of the isolator is derived using harmonic balance method, and its effectiveness is validated with numerical results. Metrics are defined to access the performance of the isolator, followed by comparison on displacement transmissibility for six types. The effect of MR damping coefficient and input amplitude on the performance of the isolator is investigated. Finally, comparative study with existing isolators is conducted. Results indicate that, passive MR damping is dependent on vibration displacement, which is beneficial to suppressing peak transmissibility with a little effect at non-resonant frequencies. The results also reveal that the isolator by type 1 or 3 has broader isolation band over other types. And the SIP in type 1 has wider isolation band and lower peak transmissibility compared with existing isolators in allowable workspace. [ABSTRACT FROM AUTHOR]

Buckling of axially loaded beams is discussed in the context of Λ-fractional analysis and mechanics. An axially compressed cantilever beam is considered in the Λ-fractional space, and the critical load is defined. The variational buckling problem of the simply supported beam is considered in the Λ-fractional space. It is pointed out that the Euler–Lagrange equation corresponding to the minimization of the total energy function with the Weierstrass–Erdmann conditions is only acceptable. The Λ-fractional buckling elastic curve of a simply supported beam is presented. That elastic curve is transferred into the initial space. The post-critical buckling deformations are defined in the context of globally stable equilibrium deformations. [ABSTRACT FROM AUTHOR]

In this paper, we study the tunneling radiation from a charged-accelerating AdS black hole with gauge potential under the impact of quantum gravity. Using the semi-classical phenomenon known as the Hamilton–Jacobi ansatz, it is studied that tunneling radiation occurs via the horizon of a black hole and also employs the Lagrangian equation using the generalized uncertainty principle. Furthermore, we investigate the impact of charge, gauge potential, and first order correction parameters on the temperature as well as the stable and unstable states of the black hole. We also compute thermodynamic properties such as entropy, internal energy, Helmholtz free energy, enthalpy, specific heat, and Gibbs free energy under the impact of the correction parameter for the black hole. We calculate the logarithmic modification terms for entropy around the equilibrium state to analyze the impacts of logarithmic correction. In the presence of the correction terms, we also check the validity of the thermodynamics. It examines the graphical representation of the influence of logarithmic correction on the thermodynamic properties of black hole stability as well as charged, accelerating, and gauge potential parameters. [ABSTRACT FROM AUTHOR]

In this work, transient source terms in advection-diffusion problems are recovered. The solution technique proposed relies on the implicit finite difference method coupled with the method of fundamental solutions, a technique for solving partial differential equations when their fundamental solution is known. The Truncated Singular Value Decomposition was used to overcome the ill-conditioned resulting linear system. The advection-diffusion inverse problem was solved using a Eulerian–Lagrangian method. Several numerical examples were considered for testing purposes by using simulated measurements perturbed with a Gaussian random noise for different Reynolds numbers. Results show that the proposed method provides stable and accurate estimates for the source term. [ABSTRACT FROM AUTHOR]

We introduce the wrapped Donaldson--Fukaya category of a (generalized) semi-toric SYZ fibration with Lagrangian section satisfying a tameness condition at infinity. Examples include the Gross fibration on the complement of an anti-canonical divisor in a toric Calabi--Yau 3-fold.We compute the wrapped Floer cohomology of a Lagrangian section and find that it is the algebra of functions on the Hori--Vafa mirror. The latter result is the key step in proving homological mirror symmetry for this case. The techniques developed here allow the construction in general of the wrapped Fukaya category on an open Calabi--Yau manifold carrying an SYZ fibration with nice behavior at infinity. We discuss the relation of this to the algebraic vs analytic aspects of mirror symmetry. [ABSTRACT FROM AUTHOR]

We study very weak solutions to scalar Euler-Lagrange equations associated with quadratic convex functionals. We investigate whether W^{1,1} solutions are necessarily W^{1,2}_{\operatorname {loc}}, which would make the theories by De Giorgi-Nash and Schauder applicable. We answer this question positively for a suitable class of functionals. This is an extension of Weyl's classical lemma for the Laplace equation to a wider class of equations under stronger regularity assumptions. Conversely, using convex integration, we show that outside this class of functionals, there exist W^{1,1} solutions of locally infinite energy to scalar Euler-Lagrange equations. In addition, we prove an intermediate result which permits the regularity of a W^{1,1} solution to be improved to W^{1,2}_{\operatorname {loc}} under suitable assumptions on the functional and solution. [ABSTRACT FROM AUTHOR]

In this paper we consider an obstacle problem for a generalization of the p-elastic energy among graphical curves with fixed ends. Taking into account that the Euler–Lagrange equation has a degeneracy, we address the question whether solutions have a flat part, i.e. an open interval where the curvature vanishes. We also investigate which is the main cause of the loss of regularity, the obstacle or the degeneracy. Moreover, we give several conditions on the obstacle that assure existence and nonexistence of solutions. The analysis can be refined in the special case of the p-elastica functional, where we obtain sharp existence results and uniqueness for symmetric minimizers. [ABSTRACT FROM AUTHOR]

For each k = 0 , ⋯ , n we construct a continuous phase f k , with f k (0) = (n - 2 k) π 2 , and viscosity sub- and supersolutions v k , u k , of the elliptic PDE ∑ i = 1 n arctan (λ i (H w)) = f k (x) such that v k - u k has an isolated maximum at the origin. It has been an open question whether the comparison principle would hold in this second order equation for arbitrary continuous phases f : R n ⊇ Ω → (- n π / 2 , n π / 2) . Our examples show it does not. [ABSTRACT FROM AUTHOR]

In the paper, we introduce a new class of noncyclic φ-contractions as a generalization of the class of noncyclic contractions which was first introduced in the paper [R. Esp'ınola, M. Gabeleh, On the structure of minimal sets of relatively nonexpansive mappings, Numerical Functional Analysis and Optimization 34 (8), 845-860, 2013] and study the existence, uniqueness and convergence of a fixed point for such class of noncyclic mapping in the framework of uniformly convex Banach spaces. We obtain existence results of the best proximity points for cyclic φ-contractions as a consequence of our main theorems. [ABSTRACT FROM AUTHOR]

The goal of this article is to compute conservation law, Lagrangian and μ-conservation law of the generalized Rosenau-type equation using the homotopy operator, the μ-symmetry method and the variational problem method. The generalized Rosenau-type equation includes the generalized Rosenau equation, the generalized Rosenau-RLW equation and the generalized Rosenau-KdV equation, which admits the third-order Lagrangian. The article also compares the conservation law and the μ-conservation law of these three equation. [ABSTRACT FROM AUTHOR]

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The classical Lagrange problem for dynamical systems introduces a Lagrangian action functional defined for any dynamical process that is envisioned to take place over a fixed interval of time with its state at each time lying on an unknown, but prescribed, configuration between two given end points in an n -dimensional state space R n . It is proposed that the fundamental dynamical field equation that characterizes the dynamical process and determines the precise motion between the two given end points is the Euler–Lagrange equation related to the stationarity of the Lagrangian action functional, expressed as the integral of a particularly formulated action density over the fixed time interval, among all admissible configurations that span the two given end points. Thus stated, this variational calculus problem introduces variations of a configuration that carries a dynamical process, and emphasizes the novelty and need to express explicitly how the configuration influences the state of that process. At each time during a dynamical process the state is subjected to an extrinsic force (classically taken to be conservative) which must be transmitted to the configuration that carries the process and, by action-reaction the configuration responds with a configuration contact force on the state of equal magnitude but opposite direction. This allows the Lagrangian action functional for a dynamical process to be interpreted as the difference between the average kinetic energy of the dynamical process that is carried by that configuration and the average configurational work done by the configuration contact force on the moving state as the state traverses that configuration during the fixed time interval. The aim in the Problem of Lagrange is to extremize this difference over all admissible configurations. The implication is that given a time interval and initial and final end points in the space of all states, the dynamical process of physical interest must follow a configuration that optimizes the gap between the average expended kinetic energy and the average expended configurational work. When the optimal condition is met and the dynamical process is so restricted, the difference between these average expenditures of energy and work will be at a local maximum, a local minimum, or a saddle point known as a condition of "least action". Herein, we investigate the optimization implications of this novel interpretation of the action functional for the Problem of Lagrange for dynamical systems for a general, possibly non-conservative, state-dependent extrinsic force field. We show that only a conservative state-dependent extrinsic force field is allowable within the statement of the problem and, thus, reaffirm the predominant classical hypothesis of restricting attention to conservative extrinsic force fields. We close with a section which covers an analogous investigation of the Problem of Lagrange for nonlinear elastodynamics. [ABSTRACT FROM AUTHOR]

A new fractional strain tensor ϵ α (u) of order α (0 < α < 1 ) is introduced for a displacement u of a body occupying the entire three-dimensional space. For α ↑ 1 , the fractional strain tensor approaches the classical infinitesimal strain tensor of the linear elasticity. It is shown that ϵ α (u) satisfies Korn's inequality (in a general L p version, 1 < p < ∞ ) and the fractional analog of Saint-Venant's compatibility condition. The strain ϵ α (u) is then used to formulate a three-dimensional fractional linear elasticity theory. The equilibrium of the body in an external force f is determined by the Euler-Lagrange equation of the total energy functional. The solution u is given by Green's function G α : u (x) = ∫ R n G α (x − y) f (y) d y , x ∈ R 3. For an isotropic body the equilibrium equation reads − μ (− Δ) α u + (λ + μ) ∇ α div α u + f = 0 where λ , μ are the Lamé moduli of the material and (− Δ) α , ∇ α and div α are the fractional laplacean, gradient and divergence. Green's function can be determined explicitly in this case: G α (x) = c α μ | x | 3 − 2 α (I − λ + μ 2 α (λ + 2 μ) (I + (2 α − 3) | x | − 2 x ⊗ x)) , x ∈ R 3 , x ≠ 0 , where I is the identity tensor (matrix), and c α a normalization factor (determined below). For α ↑ 1 the function G α approaches Green's function of the standard linear elasticity. Similar approach applies to the equilibrium solution. [ABSTRACT FROM AUTHOR]

We explicitly characterize boundary conditions that are compatible with low order variational principles. The freedom afforded by adding in a null Lagrangian without altering the Euler–Lagrange equation significantly expands the range of variationally admissible boundary conditions, although not all possibilities are permitted. Applications to several fundamental problems arising in elastostatics, including bars, beams, and plates, are presented. [ABSTRACT FROM AUTHOR]

In all Global Navigation Satellite Systems (GNSS) applications, the determination of the satellite orbits is an important task. In this study, we present the equations given in the Interface Specification Document of GPS and the Runge-Kutta method in the computation of the position P, velocity V, and acceleration A of the GPS satellites using the broadcast ephemeris. The definition of the differential equation describing the GPS satellite's motion has enabled us to introduce the Runge-Kutta method in the GPS orbit computation; this method uses the initial conditions determined in this study from the Keplerian elements provided in the broadcast ephemeris files. The Lagrange interpolation method is used for comparison of the results, where the vectors P, V, and A are estimated using the precise ephemeris. The difference not exceeding 2.4 m was obtained in the X, Y, and Z axes during seven days on the position of the GPS satellite number 9 tested in this study. In velocity and acceleration, the difference is about a few mm/s and mm/s2, respectively. [ABSTRACT FROM AUTHOR]

This study presents a novel design for a tri-stable piezoelectric vibration energy harvester with an asymmetric structure, which is enhanced with an elastic base (TPVEH + EB), meticulously designed to enhance energy extraction from irregular vibrations in architectural structures. The cornerstone of this design is the asymmetric tri-stable piezoelectric cantilever beam, distinctively arranged within a U-shaped block and fortified with an elastic foundation. A carefully positioned spring (kf)-mass (Mf) system between the U-shaped block and the beam's fixed end significantly boosts the vertical displacement of the beam during oscillations. Utilizing Lagrange's equations, we formulated a dynamic model for the asymmetric TPVEH + EB, examining the effects of potential well asymmetry, the stiffness of the elastic base and spring-mass system, the mass of the spring-mass system, and the tip magnet mass on the system's nonlinear dynamic responses. Our results demonstrate that the asymmetric TPVEH + EB significantly enhances energy harvesting from low-amplitude random vibrations (1.5 g), with the output voltage of the asymmetric TPVEH + EB increasing by 30% and the output power by 25%. Extensive numerical and theoretical analyses verify that the asymmetric TPVEH + EB provides a highly efficient solution for scenarios typically hindered by low energy conversion rates. Its reliable performance under varied and unpredictable excitation conditions highlights its excellence in advanced energy harvesting applications. The improvements detailed in this research underscore the potential of the asymmetric TPVEH + EB to boost energy harvesting efficiency, particularly in powering wireless sensor nodes for structural health monitoring in buildings. By overcoming the limitations of traditional harvesters, the asymmetric TPVEH + EB ensures enhanced efficiency and reliability, making it an ideal solution for a wide range of practical applications in diverse environmental conditions within buildings. [ABSTRACT FROM AUTHOR]

Trajectory optimization is a complex process that includes an infinite number of possibilities and combinations. This work focuses on a particular aspect of the trajectory optimization, related to the optimization of a velocity along a predefined path, with the aim of minimizing power consumption. To tackle the problem, a functional formulation and minimization strategy is developed, by means of the Euler-Lagrange equation. The minimization is later performed using a neural network approach. The strategy is deemed Lagrange-Net, as it is based on the minimization of the energy functional, by the means of Lagrange's equation and neural network approximations. [ABSTRACT FROM AUTHOR]

In this paper, we present a new asymmetrical under-actuated micro-gripper which can perform twisting and gripping operation to the target simultaneously actuated by a single piezoelectric actuator. Two improved hybrid amplification mechanisms were designed integrated with three different flexure hinges to enhance dynamic performances. Kinematics and dynamics models of the micro-gripper including input stiffness, displacement amplification ratio, and natural frequency based on pseudo-rigid-body method and Lagrange's equations were derived. Proposed models were evaluated by finite element simulation studies. Experimental results shown that our designed micro-gripper possesses good performance in terms of clamping reliability and dynamic response. [ABSTRACT FROM AUTHOR]

The variational iteration method, a powerful tool for solving nonlinear oscillators with initial conditions, has been expanded to accommodate two-point boundary conditions. In this modification, two Lagrange multipliers are introduced, and their identification process mirrors that of the conventional approach. A generalized equation is provided for a category of highly nonlinear mechanical systems, followed by three illustrative examples derived from this generalized equation. These examples serve to demonstrate the effectiveness of the method. The solutions produced by this modified approach not only show remarkable agreement with numerical results but also exhibit superior accuracy when compared to outcomes obtained through other established methods. [ABSTRACT FROM AUTHOR]

A finite element based computational scheme is developed and employed to assess a duality based variational approach to the solution of the linear heat and transport PDE in one space dimension and time, and the nonlinear system of ODEs of Euler for the rotation of a rigid body about a fixed point. The formulation turns initial-(boundary) value problems into degenerate elliptic boundary value problems in (space)-time domains representing the Euler-Lagrange equations of suitably designed dual functionals in each of the above problems. We demonstrate reasonable success in approximating solutions of this range of parabolic, hyperbolic, and ODE primal problems, which includes energy dissipation as well as conservation, by a unified dual strategy lending itself to a variational formulation. The scheme naturally associates a family of dual solutions to a unique primal solution; such 'gauge invariance' is demonstrated in our computed solutions of the heat and transport equations, including the case of a transient dual solution corresponding to a steady primal solution of the heat equation. Primal evolution problems with causality are shown to be correctly approximated by noncausal dual problems. [ABSTRACT FROM AUTHOR]

Successive drops of coloured ink mixed with surfactant are deposited onto a thin film of water to create marbling patterns in the Japanese art technique of Suminagashi. To understand the physics behind this and other applications where surfactant transports adsorbed passive matter at gas–liquid interfaces, we investigate the Lagrangian trajectories of material particles on the surface of a thin film of a confined viscous liquid under Marangoni-driven spreading by an insoluble surfactant. We study a model problem in which several deposits of exogenous surfactant simultaneously spread on a bounded rectangular surface containing a pre-existing endogenous surfactant. We derive Eulerian and Lagrangian formulations of the equations governing the Marangoni-driven surface flow. Both descriptions show how confinement can induce drift and flow reversal during spreading. The Lagrangian formulation captures trajectories without the need to calculate surfactant concentrations; however, concentrations can still be inferred from the Jacobian of the map from initial to current particle position. We explore a link between thin-film surfactant dynamics and optimal transport theory to find the approximate equilibrium locations of material particles for any given initial condition by solving a Monge–Ampère equation. We find that as the endogenous surfactant concentration $\delta$ vanishes, the equilibrium shapes of deposits using the Monge–Ampère approximation approach polygons with corners curving in a self-similar manner over lengths scaling as $\delta ^{1/2}$. We explore how Suminagashi patterns may be produced by using computationally efficient successive solutions of the Monge–Ampère equation. [ABSTRACT FROM AUTHOR]