Search Results (8)

The buckling behavior of rectangular thin plates, which are supported at their four corner points with four edges free, is a matter of great concern in the field of plate and shell mechanics. Nevertheless, the complexities arising from the boundary conditions and governing equations present a formidable obstacle to the attainment of analytical solutions for these problems. Despite the availability of various approximate/numerical methods for addressing these challenges, the literature lacks accurate analytic solutions. In this study, we employ the symplectic superposition method, a recently developed method, to effectively analyze the buckling problem of rectangular thin plates analytically. These plates have four supported corners and four free edges. To achieve this, the problem is divided into two sub-problems and solve them separately using variable separation and symplectic eigen expansion, leading to analytical solutions. Finally, we obtain the resolution to the initial issue by superposing the sub-problems. The current solution method can be regarded as a logical, analytical, and rational approach as it begins with the basic governing equation and is systematically derived without assuming the forms of the solutions. To examine various aspect ratios and in-plane load ratios of rectangular thin plates, which are supported at their four corner points with four edges free, we provide numerical examples that demonstrate the buckling loads and typical buckling mode shapes. Full article

The Jordan–Schwinger map allows us to go from a matrix representation of any arbitrary Lie algebra to an oscillator (bosonic) representation. We show that any Lie algebra can be considered for this map by expressing the algebra generators in terms of the oscillator creation and annihilation operators acting in the Hilbert space of quantum oscillator states. Then, to describe quantum states in the probability representation of quantum oscillator states, we express their density operators in terms of conditional probability distributions (symplectic tomograms) or Husimi-like probability distributions. We illustrate this general scheme by examples of qubit states (spin-1/2 $su\left(2\right)$-group states) and even and odd Schrödinger cat states related to the other representation of $su\left(2\right)$-algebra (spin-j representation). The two-mode coherent-state superpositions associated with cyclic groups are studied, using the Jordan–Schwinger map. This map allows us to visualize and compare different properties of the mentioned states. For this, the $su\left(2\right)$ coherent states for different angular momenta j are used to define a Husimi-like Q representation. Some properties of these states are explicitly presented for the cyclic groups $C_2$ and $C_3$. Also, their use in quantum information and computing is mentioned. Full article

Semiconductor chips on a substrate have a wide range of applications in electronic devices. However, environmental temperature changes may cause mechanical buckling of the chips, resulting in an urgent demand to develop analytical models to study this issue with high efficiency and accuracy such that safety designs can be sought. In this paper, the thermal buckling of chips on a substrate is considered as that of plates on a Winkler elastic foundation and is studied by the symplectic superposition method (SSM) within the symplectic space-based Hamiltonian system. The solution procedure starts by converting the original problem into two subproblems, which are solved by using the separation of variables and the symplectic eigenvector expansion. Through the equivalence between the original problem and the superposition of subproblems, the final analytical thermal buckling solutions are obtained. The SSM does not require any assumptions of solution forms, which is a distinctive advantage compared with traditional analytical methods. Comprehensive numerical results by the SSM for both buckling temperatures and mode shapes are presented and are well validated through comparison with those using the finite element method. With the solutions obtained, the effects of the moduli of elastic foundations and geometric parameters on critical buckling temperatures and buckling mode shapes are investigated. Full article

In a flexible electronic heater (FEH), periodic metal wires are often encapsulated into the soft elastic substrate as heat sources. It is of great significance to develop analytic models on transient heat conduction of such an FEH in order to provide a rapid analysis and preliminary designs based on a rapid parameter analysis. In this study, an analytic model of transient heat conduction for bi-layered FEHs is proposed, which is solved by a novel symplectic superposition method (SSM). In the Laplace transform domain, the Hamiltonian system-based governing equation for transient heat conduction is introduced, and the mathematical techniques incorporating the separation of variables and symplectic eigen expansion are manipulated to yield the temperature solutions of two subproblems, which is followed by superposition for the temperature solution of the general problem. The Laplace inversion gives the eventual temperature solution in the time domain. Comprehensive time-dependent temperatures by the SSM are presented in tables and figures for benchmark use, which agree well with their counterparts by the finite element method. A parameter analysis on the influence of the thermal conductivity ratio is also studied. The exceptional merit of the SSM is on a direct rigorous derivation without any assumption/predetermination of solution forms, and thus, the method may be extended to more heat conduction problems of FEHs with more complex structures. Full article

Entropic dynamics (ED) are a general framework for constructing indeterministic dynamical models based on entropic methods. ED have been used to derive or reconstruct both non-relativistic quantum mechanics and quantum field theory in curved space-time. Here we propose a model for a quantum scalar field propagating in dynamical space-time. The approach rests on a few key ingredients: (1) Rather than modelling the dynamics of the fields, ED models the dynamics of their probabilities. (2) In accordance with the standard entropic methods of inference, the dynamics are dictated by information encoded in constraints. (3) The choice of the physically relevant constraints is dictated by principles of symmetry and invariance. The first of such principle imposes the preservation of a symplectic structure which leads to a Hamiltonian formalism with its attendant Poisson brackets and action principle. The second symmetry principle is foliation invariance, which, following earlier work by Hojman, Kuchař, and Teitelboim, is implemented as a requirement of path independence. The result is a hybrid ED model that approaches quantum field theory in one limit and classical general relativity in another, but is not fully described by either. A particularly significant prediction of this ED model is that the coupling of quantum fields to gravity implies violations of the quantum superposition principle. Full article

This paper provides a geometric description for Lie–Hamilton systems on ${\mathbb{R}}^2$ with locally transitive Vessiot–Guldberg Lie algebras through two types of geometric models. The first one is the restriction of a class of Lie–Hamilton systems on the dual of a Lie algebra to even-dimensional symplectic leaves relative to the Kirillov-Kostant-Souriau bracket. The second is a projection onto a quotient space of an automorphic Lie–Hamilton system relative to a naturally defined Poisson structure or, more generally, an automorphic Lie system with a compatible bivector field. These models give a natural framework for the analysis of Lie–Hamilton systems on ${\mathbb{R}}^2$ while retrieving known results in a natural manner. Our methods may be extended to study Lie–Hamilton systems on higher-dimensional manifolds and provide new approaches to Lie systems admitting compatible geometric structures. Full article

Entropic Dynamics (ED) is a framework in which Quantum Mechanics is derived as an application of entropic methods of inference. In ED the dynamics of the probability distribution is driven by entropy subject to constraints that are codified into a quantity later identified as the phase of the wave function. The central challenge is to specify how those constraints are themselves updated. In this paper we review and extend the ED framework in several directions. A new version of ED is introduced in which particles follow smooth differentiable Brownian trajectories (as opposed to non-differentiable Brownian paths). To construct ED we make use of the fact that the space of probabilities and phases has a natural symplectic structure (i.e., it is a phase space with Hamiltonian flows and Poisson brackets). Then, using an argument based on information geometry, a metric structure is introduced. It is shown that the ED that preserves the symplectic and metric structures—which is a Hamilton-Killing flow in phase space—is the linear Schrödinger equation. These developments allow us to discuss why wave functions are complex and the connections between the superposition principle, the single-valuedness of wave functions, and the quantization of electric charges. Finally, it is observed that Hilbert spaces are not necessary ingredients in this construction. They are a clever but merely optional trick that turns out to be convenient for practical calculations. Full article

The basic notion of physical system states is different in classical statistical mechanics and in quantum mechanics. In classical mechanics, the particle system state is determined by its position and momentum; in the case of fluctuations, due to the motion in environment, it is determined by the probability density in the particle phase space. In quantum mechanics, the particle state is determined either by the wave function (state vector in the Hilbert space) or by the density operator. Recently, the tomographic-probability representation of quantum states was proposed, where the quantum system states were identified with fair probability distributions (tomograms). In view of the probability-distribution formalism of quantum mechanics, we formulate the superposition principle of wave functions as interference of qubit states expressed in terms of the nonlinear addition rule for the probabilities identified with the states. Additionally, we formulate the probability given by Born’s rule in terms of symplectic tomographic probability distribution determining the photon states. Full article