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We carry on the approach used in [L. Schiavone, The inverse problem within free Electrodynamics and the coisotropic embedding theorem, Inter. J. Geom. Meth. Mod. Phys.21(7) (2024) 2450131] to provide a solution for the inverse problem of the calculus of variations for Maxwell equations in vacuum and we provide an abstract theory including all implicit differential equations that can be formulated in terms of vector fields over pre-symplectic manifolds. [ABSTRACT FROM AUTHOR]

A covariant, global, variational framework for perturbations in field theories is presented. Perturbations are obtained as vertical vector fields on the configuration bundle and they drag, exactly, solution into solutions. The flow of a perturbation drags solutions into solutions and the dragged perturbed solutions can be expanded in a series with respect to the flow parameter, hence it contains perturbations at any order. Mechanics is included as a special case. As a simple application, we recover the well-known discussion about stability of geodesics on a sphere S 2 . [ABSTRACT FROM AUTHOR]

In this paper we consider optimal control problems where the control variable is a potential and the state equation is an elliptic partial differential equation of Schrödinger type, governed by the Laplace operator. The cost functional involves the solution of the state equation and a penalization term for the control variable. While the existence of an optimal solution simply follows by the direct methods of the calculus of variations, the regularity of the optimal potential is a difficult question and under the general assumptions we consider, no better regularity than the BV \mathrm{BV} one can be expected. This happens in particular for the cases in which a bang-bang solution occurs, where optimal potentials are characteristic functions of a domain. We prove the BV \mathrm{BV} regularity of optimal solutions through a regularity result for PDEs. Some numerical simulations show the behavior of optimal potentials in some particular cases. [ABSTRACT FROM AUTHOR]

This paper investigates the variational approach using nabla (denoted as ∇) within the framework of time scales. By employing two different methods, we derive the Euler-Lagrange equations for first-order variational approach to optimization problems involving exponential functions, as well as for those with both exponential functions and their ∇-derivatives. To establish the high-order variational approach to optimization problem, we present the Leibniz Formula for ∇-derivatives along with its proof. Additionally, we determine the high-order variational approach to optimization problem incorporating ∇-derivatives of exponential functions. Through these analyses, we aim to contribute to the understanding and application of the variational calculus on time scales, offering insights into the behavior of dynamic systems governed by exponential functions and their derivatives. [ABSTRACT FROM AUTHOR]

Loss functions are an essential part in modern data-driven approaches, such as bi-level training scheme and machine learnings. In this paper, we propose a loss function consisting of a r -order (an)-isotropic total variation semi-norms TV r , r ∈ ℝ + , defined via the Riemann–Liouville (RL) fractional derivative. We focus on studying key theoretical properties, such as the lower semi-continuity and compactness with respect to both the function and the order of derivative r , of such loss functions. [ABSTRACT FROM AUTHOR]

The paper aims to propose a two-step iterated method, which is derived from a regularized dynamical system of extragradient-type in terms of time discretizing, for solving an equilibrium problem. We prove that the iterative sequence generated by the method converges strongly to a solution of the equilibrium problem. Some numerical experiments are given to illustrate and compare the behavior of the new method with several other methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a forward-backward splitting dynamical system designed to address the inclusion problem of the form 0 ∈ G (x) + F(x), where G is a multi-valued operator and F is a single-valued operator in Hilbert spaces. The involved operators are required to satisfy a generalized monotonicity condition, which is less restrictive than standard monotone assumptions. Also, the maximality property does not impose on our involved operators. With mild conditions on parameters, we demonstrate the finite-time stability of the proposed dynamical system. We also present some applications to other optimization problems, such as Constrained Optimization Problems (COPs), Mixed Variational Inequalities (MVIs), and Variational Inequalities (VIs). [ABSTRACT FROM AUTHOR]

Through suitable changes of variables for a typical problem in hyper-elasticity, either in the reference or deformed configurations, one can setup and analyze versions of the same problem in terms of inner or outer maps or variations. Though such kind of transformations are part of the classical background in the Calculus of Variations, we explore under what sets of hypotheses such versions can be shown to have minimizers and be equivalent to the standard form of the problem. Such sets of hypotheses lead naturally to some distinct poly-convex energy densities for hyper-elasticity. Likewise we explore optimality in either of the two forms through a special way to generate one-parameter families of feasible deformations, feasibility including injectivity and non-interpenetration of matter. [ABSTRACT FROM AUTHOR]

A new scheme for hyperelastic material is developed based on applying the argument of calculus variation to two-factor multiplicative decomposition of the deformation gradient. Then, Piola–Kirchhoff stress is coupled with internal balance equation. Strain energy function is expressed in terms of principal invariants of the deformation gradient decomposed counterparts. Recent work introduces a strain energy function in terms of principal stretches of the deformation gradient multiplicatively decomposed counterparts directly. Hence, a new reformulation of Piola–Kirchhoff stress and internal balance equation are provided. This work focuses on developing the mathematical framework to calculate the elasticity tensor for material model formulated in terms of decomposed principal stretches. This paves the way for future implementation of these classes of material model in FE formulation. [ABSTRACT FROM AUTHOR]

It is known that a simply connected Riemann surface satisfies the isoperimetric equality if and only if it has constant Gaussian curvature. In this paper, we show that the circles centered at origin in the Randers Poincaré disc satisfy the isoperimetric equality with respect to different volume forms however, these Randers metrics do not necessarily have constant (negative) flag curvature. In particular, we show that Osserman’s result [12] of the Riemannian case cannot be extended to the Finsler geometry as such. [ABSTRACT FROM AUTHOR]

The history of variational calculus dates back to the late 17th century when Johann Bernoulli presented his famous problem concerning the brachistochrone curve. Since then, variational calculus has developed intensively as many problems in physics and engineering are described by equations from this branch of mathematical analysis. This paper presents two non-classical, distinct methods for solving such problems. The first method is based on the differential transform method (DTM), which seeks an analytical solution in the form of a certain functional series. The second method, on the other hand, is based on the physics-informed neural network (PINN), where artificial intelligence in the form of a neural network is used to solve the differential equation. In addition to describing both methods, this paper also presents numerical examples along with a comparison of the obtained results.Comparingthe two methods, DTM produced marginally more accurate results than PINNs. While PINNs exhibited slightly higher errors, their performance remained commendable. The key strengths of neural networks are their adaptability and ease of implementation. Both approaches discussed in the article are effective for addressing the examined problems. [ABSTRACT FROM AUTHOR]

In the context of quantum symmetric calculus, this study proposed more refined version of Ostrowski and Hermite–Hadamard type inequalities. The function involved in these inequalities are convex functions. In order to reach the target, left and right quantum symmetric derivative and corresponding integral are used. Furthermore, the Hölder inequality is established in the frame work of left and right quantum symmetric integral. The new results refined the results about integral inequalities that exist in the literature. [ABSTRACT FROM AUTHOR]

We describe a "discretize-then-relax" strategy to globally minimize integral functionals over functions u in a So bolev space subject to Dirichlet boundary conditions. The strategy applies whenever the integral functional depends polynomially on u and its derivatives, even if it is nonconvex. The "discretize" step uses a bounded finite element scheme to approximate the integral minimization problem with a convergent hierarchy of polynomial optimization problems over a compact feasible set, indexed by the decreasing size h of the finite element mesh. The "relax" step employs sparse moment-sum-of-squares relaxations to approximate each polynomial optimization problem with a hierarchy of convex semidefinite programs, indexed by an increasing relaxation order w. We prove that, as w → ∞ and h → 0, solutions of such semidefinite programs provide approximate minimizers that converge in a suitable sense (including in certain Lp norms) to the global minimizer of the original integral functional if it is unique. We also report computational experiments showing that our numerical strategy works well even when technical conditions required by our theoretical analysis are not satisfied. [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 low-thrust fuel-optimal transfer problem between periodic orbits of two circular restricted three-body systems has been considered. The focus is on utilizing invariant manifolds of the respective systems. A power-limited, variable-specific impulse engine model has been employed. The system-to-system transfer problem is modeled within the framework of optimal control theory through the application of the Calculus of Variations and Pontryagin's maximum principle (PMP). Along with unknown co-states, the departure and arrival points on respective periodic orbits and manifold coast durations are chosen as design variables in the problem formulation. The solution to the associated two-point boundary value problem (TPBVP) is obtained using the shooting method. Low-thrust fuel-optimal Sun-Earth-L2 halo to Sun-Mars-L1 halo transfers are obtained. The outlined result shows a significant amount of fuel savings. Such transfer trajectories may be helpful for transferring a spacecraft or cargo mission from a near-Earth station on a periodic orbit to a similar station near Mars. • Employing a low-thrust engine model results in propellant mass saving due to their high specific impulse. • A single powered arc is required for transfers with a variable specific impulse (VSI) engine. • The system-to-system transfer problem is modeled within the framework of optimal control theory. • The goal of optimization problem is to minimize fuel consumption. • Shooting method is used to converge to the desired initial co-states of the problem. [ABSTRACT FROM AUTHOR]

We consider the anisotropic integral functional of the calculus of variations ... where ci≥0 and 2 ≤pi≤pi+1 for every i = 1 ,...n-1. We exhibit a minimizer functional, for an opportune choice of the exponents pi, which turns out to be bounded everywhere and Lipschitz continuous (or even of class C¹) in an opportune subset of Ω. [ABSTRACT FROM AUTHOR]

In this paper, we first derive Milstein schemes for an interacting particle system associated with point delay McKean–Vlasov stochastic differential equations, possibly with a drift term exhibiting super-linear growth in the state component. We prove strong convergence of order one and moment stability, making use of techniques from variational calculus on the space of probability measures with finite second-order moments. Then, we introduce an antithetic multilevel Milstein scheme, which leads to optimal complexity estimators for expected functionals of solutions to delay McKean–Vlasov equations without the need to simulate Lévy areas. [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]

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]

We address the fundamental difference between solid-solid and liquid-liquid phase transitions within the Ericksen's nonlinear elasticity paradigm. To highlight ideas, we consider the simplest nontrivial 2D problem and work with a prototypical two-phase Hadamard material which allows one to weaken the rigidity and explore the nature of solid-solid phase transitions in a "near-liquid" limit. In the language of calculus of variations we probe limits of quasiconvexity in an "almost liquid" solid by comparing the thresholds for cooperative (laminate based) and non-cooperative (inclusion based) nucleation. Using these two types of nucleation tests we obtain for our model material surprisingly tight two-sided bounds on the elastic binodal without directly computing the quasiconvex envelope. [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]

We study the calculus of variations problem in the presence of a system of differential-integral (D-I) equations. In order to identify the necessary optimality conditions for this problem, we derive the so-called D-I Euler-Lagrange equations. We also generalize this problem to other cases, such as the case of higher orders, the problem of optimal control, and we derive the so-called D-I Pontryagin equations. In special cases, these formulations lead to classical Euler-Lagrange equations. To illustrate our results, we provide simple examples and applications such as obtaining the minimum power for an RLC circuit. [ABSTRACT FROM AUTHOR]

We study the calculus of variations problem in the presence of a system of differential-integral (D-I) equations. In order to identify the necessary optimality conditions for this problem, we derive the so-called D-I Euler–Lagrange equations. We also generalize this problem to other cases, such as the case of higher orders, the problem of optimal control, and we derive the so-called D-I Pontryagin equations. In special cases, these formulations lead to classical Euler–Lagrange equations. To illustrate our results, we provide simple examples and applications such as obtaining the minimumpower for an RLC circuit.

We introduce topological conditions on a broad class of functionals that ensure that the persistent homology modules of their associated sublevel set filtration admit persistence diagrams, which, in particular, implies that they satisfy generalized Morse inequalities. We illustrate the applicability of these results by recasting the original proof of the Unstable Minimal Surface Theorem given by Morse and Tompkins in a modern and rigorous framework. [ABSTRACT FROM AUTHOR]

We show the existence of a weak solution for a system of partial differential equations describing the motion of a flexible solid inside a fluid: A nonlinear, viscoelastic, n-dimensional bulk solid governed by a PDE including inertia is interacting with an incompressible fluid governed by the (n-dimensional) Navier-Stokes equation for n ≥ 2. The result is the first allowing for large bulk deformations in the regime of long time existence for fluid-structure interactions. The existence is achieved by introducing a novel variational scheme involving two time-scales that allows us to extend the method of minimizing movements to hyperbolic problems involving nonconvex and degenerate energies. [ABSTRACT FROM AUTHOR]

In this paper, we show the existence of a steady-state solution $ [\rho_0 $, $ u_0\equiv 0 $, $ \Phi_0] $ of the three-dimensional Euler-Poisson-Korteweg system in a bounded domain with physical boundary conditions using calculus of variations. Based on the existence result, the asymptotic stability for the Euler-Poisson-Korteweg system is established near the given steady state. [ABSTRACT FROM AUTHOR]

Using a modified version of Weinstein's argument for constrained minimization in nonlinear dispersive equations, we prove the existence of solitary waves in fully nonlocally nonlinear equations, as long as the linear multiplier is of positive and slightly higher order than the Coifman–Meyer nonlinear multiplier. It is therefore the relative order of the linear term over the nonlinear one that determines the method and existence for these types of equations. In analogy to Korteweg–De Vries-type equations and water waves in the capillary regime, smooth solutions of all amplitudes can be found. We consider two structural types of symmetric Coifman–Meyer symbols $ n(\xi-\eta,\eta) $, and show that cyclical symmetry is necessary for the existence of a functional formulation. Estimates for the solution and wave speed are given as the solutions tend to the bifurcation point of solitary waves. [ABSTRACT FROM AUTHOR]

Variational principles are very important for a lot of nonlinear problems to be analyzed theoretically or solved numerically. By the popular semi-inverse method and designing trial-Lagrange functionals skillfully, new variational principles are constructed successfully for the Kuramoto-Sivashinsky equation and the Coupled KdV equations, respectively, which can model a lot of nonlinear waves in shallow water. The established variational principles are also proved correct. The procedure reveals that the used technologies are very powerful and applicable, and can be extended to other nonlinear physical and mathematical models.

In this paper, we present the coisotropic embedding theorem as a tool to provide a solution for the inverse problem of the calculus of variations for a particular class of implicit differential equations, namely the equations of motion of free Electrodynamics. [ABSTRACT FROM AUTHOR]

A mean-field linear quadratic stochastic (MF-SLQ for short) optimal control problem with hybrid disturbances and cross terms in a finite horizon is concerned. The state equation is a systems driven by the Wiener process and the Poisson random martingale measure disturbed by some stochastic perturbations. The cost functional is also disturbed, which means more general cases could be characterized, especially when extra environment perturbations exist. In this paper, the well-posedness result on the jump diffusion systems is obtained by the fixed point theorem and also the solvability of the MF-SLQ problem. Actually, by virtue of adjoint variables, classic variational calculus, and some dual representation, an optimal condition is derived. Throughout our research, in order to connect the optimal control and the state directly, two Riccati differential equations, a BSDE with random jumps and an ordinary equation (ODE for short) on disturbance terms are obtained by a decoupling technique, which provide an optimal feedback regulator. Meanwhile, the relationship between the two Riccati equations and the so-called mean-field stochastic Hamilton system is established. Consequently, the optimal value is characterized by the initial state, disturbances, and original value of the Riccati equations. Finally, an example is provided to illustrate our theoretic results. [ABSTRACT FROM AUTHOR]

We study a family of representations of the canonical commutation relations (CCR)-algebra, which we refer to as "admissible," with an infinite number of degrees of freedom. We establish a direct correlation between each admissible representation and a corresponding Gaussian stochastic calculus. Moreover, we derive the operators of Malliavin's calculus of variation using an algebraic approach, which differs from the conventional methods. The Fock-vacuum representation leads to a maximal symmetric pair. This duality perspective offers the added advantage of resolving issues related to unbounded operators and dense domains much more easily than with alternative approaches. [ABSTRACT FROM AUTHOR]

The unrestrained torsional analysis of bars is an important theme in elasticity theory, first solved by Saint-Venant using semi-inverse methods. It has been considered and solved by several others using analytical methods and numerical procedures due to the importance in the design of machine parts under torsional moments. In this paper, the Saint Venant torsion problem is solved for rectangular prismatic bars using Galerkin-Kantorovich variational method (GKVM). The work presents a detailed theoretical framework of the problem, deriving using first principles considerations the stress compatibility equation in terms of the Prandtl stress function ϕ(x, y). The derived domain equation which is required to be satisfied over the rectangular cross-sectional domain is a partial differential equation of the Poisson type. GKVM is adopted as the solution method for finding the solution to the domain equation. The unknown Prandtl stress function ϕ(x, y) is assumed, following Kantorovich method to be a product of an unknown function for f(x) sought to minimize the Galerkin-Kantorovich variational functional (integral) (GKVF) and a known function (y2 - n2) which satisfies the boundary conditions at all boundary points in the y-direction, that is, at y = ±b. The resulting GKVF is a simplified functional whose integral is a second order inhomogeneous ordinary differential equation (ODE) in f(x). The integrand is solved to find f(x) leading in a full determination of the Prandtl stress function. The expression for stresses, torsional moments and torsional parameters are then found and they satisfy the boundary conditions and the domain equation. The results for the torsional moments and torsional parameters are identical to previous results obtained using double finite sine transform method (DFSTM), and analytical methods. The merit of GKVM is that it has led to the exact solution of the unrestrained torsion problems. [ABSTRACT FROM AUTHOR]

We study Γ-convergence of nonconvex integrals of the calculus of variations in strongly connected sets when the integrands do not have polynomial growth and can take infinite values. Applications to homogenization of unbounded integrals in strongly perforated sets are also developed. [ABSTRACT FROM AUTHOR]

In this paper, we present the general Bernstein functions for the first time. The properties of generalized Bernstein basis functions are given and demonstrated. The classical Bernstein polynomial bases are merely a subset of the general Bernstein functions. Based on the new Bernstein base functions and the collocation method, we present a numerical method for solving linear and nonlinear χ-fractional differential equations (χ-FDEs) with variable coefficients. The fractional derivative used in this work is the χ-Caputo fractional derivative sense (χ-CFD). Combining the Bernstein functions basis and the collocation methods yields the approximation solution of nonlinear differential equations. These base functions can be used to solve many problems in applied mathematics, including calculus of variations, differential equations, optimal control, and integral equations. Furthermore, the convergence of the method is rigorously justified and supported by numerical experiments. [ABSTRACT FROM AUTHOR]

This article studies the pullback dynamics and statistical solutions for the dissipative non-autonomous Zakharov equations in a bounded interval. The main obstacle in our investigating comes from the geometric constrain of global regularity of solutions for the Zakharov equations. This geometric constrain is caused essentially by its particular structure which is a special coupling of a hyperbolic equation with a dispersive nonlinear Schrödinger equation. The nonlinear term Δ | E | 2 in the hyperbolic equation is the core ingredient leading to this geometric constrain. Firstly we show the global existence and uniqueness of solutions, and prove the continuous dependence of solutions on the initial data by exploiting a certain coercivity of the operator corresponding to the linear principle part of the Zakharov equations. Then we establish that the generated process of solution mappings possesses a bounded pullback absorbing set and pullback asymptotic compactness by using a delicate decomposition of the original problem, and obtain the existence of a pullback attractor. Next we formulate an appropriate definition of τ -continuity for the process and employ the calculus of variations to prove this τ -continuity. Afterwards, we construct a family of invariant Borel probability measures for the process via the pullback attractor and the notion of generalized Banach limit. We also propose an appropriate definition of statistical solutions for the addressed Zakharov equations, and establish that the constructed family of invariant Borel probability measures is indeed a statistical solution which satisfies the Liouville theorem of Statistical Mechanics. Our definitions and approach present here can overcome effectively the difficulty caused by the geometric constrain of the Zakharov equations. Finally, we propose some open problems related to the existence and singular limiting behavior of the pullback attractors and statistical solutions for the Zakharov equations with varying coefficient. [ABSTRACT FROM AUTHOR]

In this study, the goal is to obtain optimal solutions for fractional interval linear quadratic regulator problems. The major contribution made in this paper is to apply an indirect approach based on the interval calculus of variations with a joint application of constrained interval arithmetic to transcribe the fractional problem under study into a system of Volterra integral equations. To do so, we present the fractional interval Pontryagin's minimum principle for the extraction of necessary optimality conditions. The obtained conditions are first transformed into a system of Volterra integral equations and then using the Laplace transform method, we solve these equations and obtain the optimal solutions. One of the key points of this method is to keep the continuous form of the problems, which we can change to an equivalent form without discretizing it and proving the existence of its solution. Some numerical examples are carried out to confirm the standard criteria with theoretically accurate results. [ABSTRACT FROM AUTHOR]

This research presents a first attempt to analytically determine the asymmetric dynamic transverse characteristics of thin to moderately thick viscoelastic functionally graded porous (VFGP) annular plates. Firstly, the material properties of the plate are assumed to have various nonlinear distributions in terms of porosity coefficient. Secondly, the motion equations are obtained through the first-order shear deformation theory (FSDT) of elasticity, the energy method, and the variations calculus. Thirdly, the standard linear solid (SLS) model is adopted to consider the viscoelastic behavior of the plate. Finally, the perturbation procedure together with Fourier series are utilized to solve the system of partial differential equations, and the asymmetrically dynamic response is found in a closed-form solution. To assess the veracity of the analytical findings, an algorithm based on the finite element (FE) method called the user-defined field (USDFLD) code is developed. In order to benchmark the present study, the dynamic response of VFGP annular plates is scrutinized under two types of excitations (impulsive and step), four different types of radial load profiles (such as constant, linear, parabolic, and sine distributions), and various asymmetric circumferential loads. Moreover, the influence of various geometrical and material characteristics on the dynamic response of VFGP annular plates is investigated. [ABSTRACT FROM AUTHOR]

We investigate a dual bending energy functional that operates on the dual pseudo-sphere in dual Lorentzian space. For a nonnull dual curve on the dual pseudo-sphere to be considered elastic, it must satisfy the conditions of a dual Euler-Lagrange equation. To solve this problem, we use Jacobi elliptic functions to approach the real part and the integral factors method to solve the dual part. Using E. Study mapping, we examine situations where every timelike or spacelike dual elastic curve on the dual pseudo-sphere matches an elastic strip with a suitable base curve in Minkowski 3-space. [ABSTRACT FROM AUTHOR]

This article explores the problem of finding the fastest descent path under uniform gravity, known as the brachistochrone curve. The traditional method involves using variational calculus, but the authors propose a simpler approach using straight lines. By dividing the path into segments and calculating the time for a particle to cover each segment, a curve resembling the brachistochrone can be found. This alternative method helps students understand kinematical and dynamical concepts and introduces them to mathematical optimization. The article also discusses the properties of the cycloid curve and provides equations for calculating the descent time. The authors compare their method to the analytical solution and suggest that prior knowledge of the solution can improve accuracy. [Extracted from the article]

Generalized nonlinear programming is considered without any convexity assumption, capturing a variety of problems that include nonsmooth objectives, combinatorial structures, and set-membership nonlinear constraints. We extend the augmented Lagrangian framework to this broad problem class, pre-serving an implicit formulation and introducing auxiliary variables merely as a formal device. This, however, gives rise to a generalized augmented Lagrangian function that lacks regularity. Based on parametric optimization, we develop a tailored stationarity concept to better qualify the iterates, generated as approximate solutions to a sequence of subproblems. Using this variational characterization and the lifted representation, asymptotic properties and convergence guarantees are established for a safeguarded augmented Lagrangian scheme. Numerical examples showcase the modelling versatility gained by dropping convexity assumptions and the practical benefits of the advocated implicit approach. [ABSTRACT FROM AUTHOR]

We introduce a class of integral theorems based on cyclic functions and Riemann sums approximating integrals. The Fourier integral theorem, derived as a combination of a transform and inverse transform, arises as a special case. The integral theorems provide natural estimators of density functions via Monte Carlo methods. Assessment of the quality of the density estimators can be used to obtain optimal cyclic functions, alternatives to the sin function, which minimize square integrals. Our proof techniques rely on a variational approach in ordinary differential equations and the Cauchy residue theorem in complex analysis. [ABSTRACT FROM AUTHOR]

We exploit the existence and non-existence of positive solutions to the eigenvalue problem driven by the nonhomogeneous fractional $ p\& q $ p &q Laplacian operator with indefinite weights \[ \left(-\Delta_p\right)^{\alpha}u + \left(-\Delta_q\right)^{\beta}u = \lambda\left[a \left|u\right|^{p-2}u + b \left|u\right|^{q-2}u \right]\quad{\rm in}\ \Omega, \] (− Δ p) α u + (− Δ q) β u = λ [ a | u | p − 2 u + b | u | q − 2 u ] in Ω , where $ \Omega \subseteq \mathbb {R}^N $ Ω ⊆ R N is a smooth bounded domain that has been extended by zero. We further show the existence of a continuous family of eigenvalues in the case $ \Omega =\mathbb {R}^N $ Ω = R N and $ b\equiv 0 $ b ≡ 0 a.e. Our approach relies strongly on variational Analysis, in which the Mountain pass theorem plays the key role. Due to the lack of spatial compactness and the embedding $ \mathcal {W}^{\alpha, p}\left (\mathbb {R}^N\right) \hookrightarrow \mathcal {W}^{\beta, q}\left (\mathbb {R}^N\right) $ W α , p (R N) ↪ W β , q (R N) in $ \mathbb {R}^N $ R N , we employ the concentration-compactness principle of P.L. Lions [The concentration-compactness principle in the calculus of variations. The limit case. II, Rev Mat Iberoamericana. 1985;1(2):45–121]. to overcome the difficulty. Our paper can be considered as a counterpart to the important works [Alves et al. Existence, multiplicity and concentration for a class of fractional $ p\& q $ p &q Laplacian problems in $ \Bbb R^N $ R N , Commun Pure Appl Anal, 2019;18(4):2009–2045], [Benci et al. An eigenvalue problem for a quasilinear elliptic field equation. J Differ Equ, 2002;184(2):299–320], [Bobkov et al. On positive solutions for $ (p,q) $ (p , q) -Laplace equations with two parameters, Calc Var Partial Differ Equ, 2015;54(3):3277–3301], [Colasuonno and Squassina. Eigenvalues for double phase variational integrals, Ann Mat Pura Appl (4), 2016;195(6):1917–1956], [Papageorgiou et al. Positive solutions for nonlinear Neumann problems with singular terms and convection, J Math Pures Appl (9), 2020;136:1–21], [Papageorgiou et al. Ground state and nodal solutions for a class of double phase problems, Z Angew Math Phys, 2020;71:1–15], and may have further applications to deal with other problems. [ABSTRACT FROM AUTHOR]

Because variational principles are very important for some methods to get the numerical or exact solutions, it is very important to seek explicit variational formulations for the non-linear PDE. At first, this paper describes the modified Benjamin-Bona-Mahony equation in fractal porous media or with irregular boundaries. Then, by designing skillfully the trial-Lagrange functional, variational principles are successfully established for the modified Benjamin-Bona-Mahony equation in the fractal space, respectively. Furthermore, the obtained variational principles are proved correct by minimizing the functionals with the calculus of variations. [ABSTRACT FROM AUTHOR]

In this work, we consider fractional variational problems depending on higher order fractional derivatives. We obtain optimality conditions for such problems and we present and discuss some examples. We conclude with possible research directions. [ABSTRACT FROM AUTHOR]

This work addresses the occupation measure relaxation of calculus of variations problems, which is an infinite-dimensional linear programming reformulation amenable to numerical approximation by a hierarchy of semidefinite optimization problems. We address the problem of equivalence of this relaxation to the original problem. Our main result provides sufficient conditions for this equivalence. These conditions, revolving around the convexity of the data, are simple and apply in very general settings that may be of arbitrary dimensions and may include pointwise and integral constraints, thereby considerably strengthening the existing results. Our conditions are also extended to optimal control problems. In addition, we demonstrate how these results can be applied in nonconvex settings, showing that the occupation measure relaxation is at least as strong as the convexification using the convex envelope; in doing so, we prove that a certain weakening of the occupation measure relaxation is equivalent to the convex envelope. This opens the way to application of the occupation measure relaxation in situations where the convex envelope relaxation is known to be equivalent to the original problem, which includes problems in magnetism and elasticity. [ABSTRACT FROM AUTHOR]

We propose a systematic generating procedure to construct free Lagrangians for massive, massless and partially massless, totally-symmetric tensor fields on AdSd+1 starting from the Becchi–Rouet–Stora–Tyutin (BRST) Lagrangian description of massless fields in the flat ambient space R d , 2 . A novelty is that the Lagrangian is described by a d + 1 form on R d , 2 whose pullback to AdSd+1 gives the genuine Lagrangian defined on anti de Sitter spacetime. Our derivation uses the triplet formulation originating from the first-quantized BRST approach, where the action principle is determined by the BRST operator and the inner product of a first-quantised system. In this way we build, in a manifestly so(2, d)-covariant manner, a unifying action principle for the three types of fields mentioned above. In particular, our derivation justifies the form of some actions proposed earlier for massive and massless fields on (anti)-de Sitter. We also give a general setup for ambient Lagrangians in terms of the respective jet-bundles and variational bi-complexes. In particular we introduce a suitable ambient-space Euler–Lagrange differential which allows one to derive the equations of motion ambiently, i.e., without the need to explicitly derive the respective spacetime Lagrangian. [ABSTRACT FROM AUTHOR]

In this work, a variational principle–based approach has been adopted to analyze one of the classical linear elasticity problem of the axisymmetric cylinder under surface loading. The use of variational principle results in a set of governing partial differential equations with associated boundary conditions. The equations have been solved using the separation of variable approach and the Frobenius method. A general solution has been derived and used to solve two test cases. The proposed solution is capable of meeting all the boundary conditions. The solution has been validated by comparing it with a finite element–based numerical solution and considering a special limiting condition of a solid cylinder, for which results are available in the literature. Further various studies have been carried out to understand the robustness and limitation of the presented solution. [ABSTRACT FROM AUTHOR]

In this paper, we show that, given appropriate boundary data, the free boundaries of minimizers of functionals of type J(v; A, λ+, λ-, Ω)= ∫Ω((A(x∇v, ∇v) + ∇(v))dx and the fixed boundary touch each other in a tangential fashion. We extend the results of Karakhanyan, Kenig, and Shahgholian [Calc. Var. Partial Differential Equations 28 (2007), 15-31] to the case of variable coefficients. We prove this result via classification of the global profiles, as per Karakhanyan, Kenig, and Shahgholian [Calc. Var. Partial Differential Equations 28 (2007), 15-31]. [ABSTRACT FROM AUTHOR]

In this paper, a Euler wavelets method for solving the variational problems is presented. The operational matrices of integration and product of Euler wavelets are calculated. Then, by using Euler wavelets and the operational matrices, the variational problems are reduced into the system of algebraic equations. Furthermore, the convergence analysis and error bound of the Euler wavelets method are given. Some examples are included to demonstrate the applicability and validity of the schemes. [ABSTRACT FROM AUTHOR]

Different approaches are used to define the optimal path in terms of construction costs. Such problems in practice are usually solved by various heuristic procedures. To obtain a theoretically justified result, one can derive an integral cost functional under certain assumptions and use variational principles. Thus, the classical problem of the calculus of variations is obtained. The necessary condition for the minimum of such a functional has the form of an integrodifferential equation. This paper describes a numerical algorithm for solving this equation, which is based on the prominent shooting method, which has been studied in detail in the literature. Under additional assumptions, the existence of a solution is proved using Schauder's fixed point principle. The problem of the uniqueness of the solution is studied. A numerical example is provided. [ABSTRACT FROM AUTHOR]