Variational Principle Research Articles

UNIFICAÇÃO ELETROGRAVITACIONAL BASEADA EM PARTÍCULAS FUNDAMENTAIS E CAMPO UNIFICADOR

ABSTRACT This paper proposes a unified reformulation of the fundamental equations of physics, integrating gravitation, electromagnetism and quantum mechanics through a discrete energy field composed of elementary particles (h±, m±, l±). From this theoretical framework, new technologies can be developed, capable of levitating masses, using accessible materials and energy, exploiting density gradients of particles m±, h± and l±, according to the reformulated equations derived from the axioms listed in Appendix A. The modified Maxwell, gravitation and Schrödinger equations are derived from variational principles, and their experimental feasibility is demonstrated through computer simulations and calculations of stress, density and energy consumption. The equations and concepts are unique to this reformulation, and the results are partially validated by computer simulations. Classification IPC: - F03H — Propulsion systems not otherwise provided for. - H02K 53/00 — Motors using electromagnetic inertia manipulation. Keywords: Electrogravitation, Unified Field, Fundamen’tal Particles, Levitation, Inertial Propulsion.

Boundedness components and Ekeland's type variational principles in locally convex cones

Using the boundedness components, we extend the Ekeland's variational principle for upper, lower and symmetric relative topologies. If the weak preorder is anti-symmetric, we deduce that the corresponding set-valued functions carry fixed points on the symmetric boundedness components.

Function domains and the universal matrix functional of multi-state density functional theory.

On the basis of recent advancements in the Hamiltonian matrix density functional for multiple electronic eigenstates, this study delves into the mathematical foundation of the multistate density functional theory (MSDFT). We extend a number of physical concepts at the core of Kohn-Sham DFT, such as density representability, to the matrix density functional. In this work, we establish the existence of the universal matrix functional for many states as a proper generalization of the Lieb universal functional for the ground state. Consequently, the variation principle of MSDFT can be rigorously defined within an appropriate domain of matrix densities, thereby providing a solid framework for DFT of both the ground state and excited states. We further show that the analytical structure of the Hamiltonian matrix functional is considerably constrained by the subspace symmetry and invariance properties, requiring and ensuring that all elements of the Hamiltonian matrix functional are variationally optimized in a coherent manner until the Hamiltonian matrix within the subspace spanned by the lowest eigenstates is obtained. This work solidifies the theoretical foundation to treat multiple electronic states using density functional theory.

Fractional Optimal Control Problem for Symmetric System Involving Distributed-Order Atangana–Baleanu Derivatives with Non-Singular Kernel

The objective of this work is to discuss and thoroughly analyze the fractional variational principles of symmetric systems involving distributed-order Atangana–Baleanu derivatives. A component of distributed order, the fractional Euler–Lagrange equations of fractional Lagrangians for constrained systems are studied concerning Atangana–Baleanu derivatives. We give a general formulation and a solution technique for a class of fractional optimal control problems (FOCPs) for such systems. The dynamic constraints are defined by a collection of FDEs, and the performance index of an FOCP is considered a function of the control variables and the state. The formula for fractional integration by parts, the Lagrange multiplier, and the calculus of variations are used to obtain the Euler–Lagrange equations for the FOCPs.

Investigating the kinetics of single-chain expansion upon release in theta conditions

The free expansion of a confined chain in theta solvents following a sudden removal of the confining constraint is investigated using Langevin dynamics simulations in both two- and three-dimensional spaces. The average evolution of the chain size exhibits a sigmoidal transition between the confined and the free states on a logarithmic timescale, indicating a two-stage expansion, each characterized by its own timescale. A kinetic theory is developed by applying Onsager’s variational principle, which balances the change in free energy with energy dissipation. Through scaling analysis, the characteristic time τ1 for the first expansion stage is shown to scale as the cube of the initial chain size, while the chain size increases according to a power law with an exponent α1=1/3, independent of the spatial dimension. In the second stage, the timescale τ2 is found to be proportional to the square of the chain length, and the evolution of the chain size follows an exponential recovery function powered by an exponent α2=1/4. These results are further validated by a direct analysis of the kinetic equations via simulations. Moreover, the general forms of the free energy for the two expansion stages are established through the integration of the kinetic equations. Finally, physical interpretations are proposed, employing a radial expansion model and a diffusive mechanism to explain the observed scaling behaviors. This work explores a model system under the specific solvent condition, providing foundational theory and enhancing our understanding of the expansion-upon-release phenomenon.

Singular p(x)-Laplacian equation with application to boundary layer theory

In thispaper, we study a singular p ( x ) -Laplacian equation that incorporates both variable singular and superlinear nonlinearities. By applying Ekeland’s variational principle and a constrained minimization approach, we establish the existence and uniqueness of a positive solution when the variable singularity β ( x ) is within the interval ( 1 , ∞ ) . As an application of our results, we provide two examples from the basic problem in the boundary layer theory of these pseudoplastic fluids with the no-slip boundary condition at both plates.

Lagrangians and Newtonian analogs for biological systems

This study investigates the potential for biological systems to be governed by a variational principle, suggesting that such systems may evolve to minimize or optimize specific quantities. To explore this idea, we focus on identifying Lagrange functions that can effectively model the dynamics of selected population systems. These functions provide a deeper understanding of population evolution by framing their behavior in terms of energy-like variables. We present an algorithm for generating Lagrangian functions applicable to a family of population dynamics models and demonstrate the equivalence between two-dimensional population models and a one-dimensional Newtonian mechanical analog. Furthermore, we explore the existence of conservation laws for these models, utilizing Noether’s theorems to investigate their implications.

Real ray tracing in viscoelastic anisotropic media with homogeneous ray and energy velocities

The very recently developed g*-Hamiltonian (g*H) method has demonstrated strong potential in handling qSV waves in real ray tracing (RRT). But its investigation and application has so far been limited to homogeneous media or at an interface. In this study, the RRT equations based on the g*H method are derived and solved by the fourth-order Runge-Kutta method for four inhomogeneous viscoelastic, vertical transversely isotropic (VTI) velocity models. The ray tracing equations are solved for specified real incident slowness vectors and specified ray directions, highlighting their practical significance. However, raypaths traced for specified incident ray directions exhibit a broad shadow zone with sparsely distributed rays for the qSV wave. This phenomenon is caused by triplication (wavefront crossings) even though it is small. We also demonstrate that the g*H method can successfully trace rays even if the VTI material has large triplications on the qSV wave fronts. The calculations demonstrate the critical importance of verifying the homogeneity of ray velocities at every ray tracing step in these RRT methods, as homogeneous ray velocity vectors with parallel real and imaginary components cannot always be guaranteed. By applying the concepts of time average energy velocity and quality factor, we demonstrate that a complex energy flux velocity vector is consistently homogeneous without the need to search for stationary slowness vectors and matches the corresponding homogeneous ray velocity well. For the first time, the complex energy flux velocity vector is used for ray tracing by applying the modified Newton algorithm based on Fermat's Variational Principle.

A meshless method for thermal vibration analysis of curved-edge trapezoidal plates made of laminated materials

Abstract This paper reports a meshless method for stochastic vibration analysis of trapezoidal composite laminates with curvilinear bound. The basis functions used in this meshless method are Chebyshev polynomials, which are adapted to the defining domain of the Chebyshev polynomials by transforming the curved-edge plate into a square plate using a coordinate mapping technique. The developed vibration analysis model combines the first-order shear deformation theory (FSDT) and thermos-elasticity theory using the Hamiltonian variational principle. The displacement tolerance function of the square plate is approximated by a two-dimensional meshless Chebyshev-radial basis point interpolation method. Gaussian Lobato sampling points are used to discretize the square domain. The correctness and efficiency of the model is verified by comparison with finite element results and published literature. The effects of temperature, geometry, and stochastic loading on the vibration characteristics of curved edge trapezoidal plates are also carried out. The results show that the existing Chebyshev-radial basis point interpolation method can accurately and efficiently calculate the vibration characteristics of curved-edge trapezoidal plates under different boundary conditions.

Phase Portrait, Bifurcation and Chaotic Analysis, Variational Principle, Hamiltonian, Novel Solitary, and Periodic Wave Solutions of the New Extended Korteweg–de Vries–Type Equation

ABSTRACTThe center task of this paper is to give the qualitative and quantitative investigations into the nonlinear dynamics of the new extended Korteweg–de Vries–type equation for shallow‐water waves. Applying the traveling wave transformation and semi‐inverse method (SIM), the variational principle (VP) is developed. Based on the VP, we extract the system's Hamiltonian. The planar dynamical system is then derived using the Galilean transformation, followed by phase portrait plotting and bifurcation analysis to explore the existence of different types of wave solutions. Meanwhile, the chaotic behaviors of the system are also analyzed by taking the external perturbation terms. Eventually, two robust approaches—the variational method that stemmed from the variational principle and Ritz method—along with the Hamiltonian‐based method are employed to seek some wave solutions of the equation. Different kinds of the wave solutions like bell shape solitary, anti‐bell shape solitary, and periodic wave solutions are obtained. The findings of this exploration are all novel and help us gain a deeper understanding of the nonlinear dynamics of the equation being studied.

On the variational principles of the Burgers-Korteweg-de Vries equation in fluid mechanics

As is known to all, it is extraordinarily difficult to construct the variational principles of the nonlinear partial differential equations (NPDEs) for fluid mechanics. In this work, we focus on the Burgers-Korteweg-de Vries equation and attempt to establish its generalized variational principle by employing the semi-inverse method (SIM). Two different generalized variational principles (GVPs) are extracted and the detailed derivation process is presented. The GVPs can present some new inspiration for the study and application of the variational method.

Constructal thermodynamics and its semantic ontology in autopoietic, digital, and computational architectural and urban space open systems.

Constructal thermodynamics and its semantic ontology in autopoietic, digital, and computational architectural and urban space open systems.

Note on the Metric Entropy for Multivalued Maps

The first goal of this note is to highlight several discrepancies in the proofs of the recent paper by Vivas and Sirvent (Discrete Contin Dyn Syst Ser B 27(11):6589–6604, 2022. https://doi.org/10.3934/dcdsb.2022010). The results in question, which relate metric and topological entropies for multivalued maps and can be referred to as “half variational principles” for multivalued maps, are pioneering in this field. Our second goal is to provide revised statements accompanied by correct proofs. By employing the concept of h+ entropy for multivalued maps, developed in our earlier papers, we were able to partially strengthen the results of Vivas and Sirvent and establish the “full variational principle” for a special subclass of multivalued lower semicontinuous maps with convex compact values on a compact subset of a Banach space. However, our approach requires two rather restrictive additional assumptions, which were also used by Vivas and Sirvent in another related context.

Analysing polarisability, dipole moment, Stark shift, self-polarisation effect and diamagnetic susceptibility of GaAs quantum dot under the simultaneous influence of noise and spatial dissemination of impurity

ABSTRACT The study endeavours to unveil the role of the subtle interplay between spatial dissemination of impurity and Gaussian white noise on a few electrical and magnetic properties of doped GaAs quantum dot (QD). The properties explored are polarisability, dipole moment, Stark shift, self-polarisation effect and diamagnetic susceptibility. Linear variation principle has been used for the determination of approximate eigenstates and the energies of the system. The interplay mentioned above is delicately governed by the pathway (additive/multiplicative) by which noise makes its entry into the system and the range of spatial spread of impurity. The study reveals that, a gradual variation of impurity spread, in combination with the route of entry of noise, can noticeably tune the above properties. Under all conditions, a progressive increase in impurity spread enhances the polarisability, dipole moment, Stark shift and diamagnetic susceptibility. However, the self-polarisation effect enhances (minimises around some typical impurity spread) with an increase in the impurity spread without noise and under additive noise (multiplicative noise). Mostly, the presence of noise diminishes the spatial elongation of the electronic wave function from the noise-free situation. And the diminish enhances under multiplicative noise over its additive analogue.

A positive quasilocal mass for causal variational principles

A new inequality for a nonlinear surface layer integral is proved for minimizers of causal variational principles. This inequality is applied to obtain a new proof of the positive mass theorem with volume constraint. Next, a positive mass theorem without volume constraint is stated and proved by introducing and using the concept of asymptotic alignment. Moreover, a positive quasilocal mass and a synthetic definition of scalar curvature are introduced in the setting of causal variational principles. Our notions and results are illustrated by the explicit examples of causal fermion systems constructed in ultrastatic spacetimes and the Schwarzschild spacetime. In these examples, the correspondence to the ADM mass and similarities to the Brown–York mass are worked out.

The inverse variational principle in random dynamical systems

This paper introduces the concepts of upper and lower capacity topological pressure for non-compact sets into the context of random dynamical systems. Subsequently, we investigate their properties and establish their measurability, demonstrating that these concepts are well-defined. Finally, we derive the inverse variational principle for random dynamical systems.

Existence and multiplicity of solutions for a class Kirchhoff–Boussinesq equations with nonlocal term in ℝ N

This paper is concerned with the existence and multiplicity of nontrivial solutions for the class of elliptic Kirchhoff–Boussinesq with nonlocal term type problems given by { Δ 2 u ± Δ p u − M ( ∫ R N | ∇u | 2 d x ) Δu + V ( x ) u = f ( x ) | u | q − 2 u + g ( x ) | u | r − 2 u + | u | 2 ∗ ∗ − 2 u in R N , u ∈ H 2 ( R N ) , where Δ 2 u is the biharmonic operator, 2 < p ≤ 2 ∗ = 2 N N − 2 , Δ p u = div ( | ∇u | p − 2 ∇u ) is the p-Laplacian operator, M : R + → R + is a continuous function, f and g are functions satisfying some suitable assumptions, 2 ∗ ∗ = 2 N N − 4 is the critical Sobolev exponent such that 1 < q < 2 < p ≤ 2 ∗ < 8 N − 4 < r < 2 ∗ ∗ for 5 ≤ N ≤ 7 and 1 < q < 2 < p ≤ 2 ∗ < r < 2 ∗ ∗ for N ≥ 8 , V ( x ) is a positive potential. Unlike most other literatures on this problem, we replace Laplacian with p-Laplacian and allow its coefficient to be positive and negative. Based on the Mountain Pass Theorem and Ekeland Variational Principle, we prove such an equation possesses at least two different nontrivial solutions.

A Type II Hamiltonian Variational Principle and Adjoint Systems for Lie Groups

We present a novel Type II variational principle on the cotangent bundle of a Lie group which enforces Type II boundary conditions, i.e., fixed initial position and final momentum. In general, such Type II variational principles are only globally defined on vector spaces or locally defined on general manifolds; however, by left translation, we are able to define this variational principle globally on cotangent bundles of Lie groups. Type II boundary conditions are particularly important for adjoint sensitivity analysis, which is our motivating application. As such, we additionally discuss adjoint systems on Lie groups, their properties, and how they can be used to solve optimization problems subject to dynamics on Lie groups.

General maximum principle for optimal control problems of stochastic systems with delay: an Ekeland's variational principle method

In this paper, we consider optimal control problems of stochastic systems with delay. The control domains are not necessarily convex and the diffusion coefficients explicitly depend on control variables. The delay term of the state is regarded as a control process and then the Ekeland's variational principle is applied to obtain a general maximum principle for the optimal control problems. As applications, we use the theoretical results to a delayed stochastic linear quadratic (LQ) optimal control problem with a non-convex control domain.

Boundary terms in string field theory

We supplement the string field theory action with boundary terms to make its variational principle well-posed. Central to our considerations is the violation of the stress-energy tensor conservation in non-compact CFTs due to the boundary terms. This manifests as the failure of the cyclicity of the BRST operator, which encodes the target space integration by parts identities at the level of the worldsheet. Using this failure, we argue that the free closed string field theory action admits a well-posed variational principle upon including an additional boundary contribution. We explicitly work out the resulting action up to the massless level and show that it is related to the expansion of the low-energy effective string action endowed with the Gibbons-Hawking-York term on a flat background. We also discuss the structure of the boundary terms in the interacting theory.