From Generalized Hamilton Principle to Generalized Schrodinger Equation

The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we can obtain the standard Schrodinger equation. In this paper, we have given the generalized Hamilton principle, which can describe the heat exchange system, and the nonconservative force system. On this basis, we have further given their generalized Lagrange functions and Hamilton functions. With the Feynman path integration, we have given the generalized Schrodinger equation of nonconservative force system and the heat exchange system.

Constrained Hamilton variational principle for shallow water problems and Zu-class symplectic algorithm

In this paper, the shallow water problem is discussed. By treating the incompressible condition as the constraint, a constrained Hamilton variational principle is presented for the shallow water problem. Based on the constrained Hamilton variational principle, a shallow water equation based on displacement and pressure （SWE-DP） is developed. A hybrid numerical method combining the finite element method for spa- tial discretization and the Zu-class method for time integration is created for the SWE- DP. The correctness of the proposed SWE-DP is verified by numerical comparisons with two existing shallow water equations （SWEs）. The effectiveness of the hybrid numerical method proposed for the SWE-DP is also verified by numerical experiments. Moreover, the numerical experiments demonstrate that the Zu-class method shows excellent perfor- mance with respect to simulating the long time evolution of the shallow water.

A hybrid-stress element based on Hamilton principle

A novel hybrid-stress finite element method is proposed for constructing simple 4-node quadrilateral plane elements, and the new element is denoted as HH4-3fl here. Firstly, the theoretical basis of the traditional hybrid-stress elements, i.e., the Hellinger-Reissner variational principle, is replaced by the Hamilton variational principle, in which the number of the stress variables is reduced from 3 to 2. Secondly, three stress parameters and corresponding trial functions are introduced into the system equations. Thirdly, the displacement fields of the conventional bilinear isoparametric element are employed in the new models. Finally, from the stationary condition, the stress parameters can be expressed in terms of the displacement parameters, and thus the new element stiffness matrices can be obtained. Since the required number of stress variables in the Hamilton variational principle is less than that in the Hellinger-Reissner variational principle, and no additional incompatible displacement modes are considered, the new hybrid-stress element is simpler than the traditional ones. Furthermore, in order to improve the accuracy of the stress solutions, two enhanced post-processing schemes are also proposed for element HH4-3β. Numerical examples show that the proposed model exhibits great improvements in both displacement and stress solutions, implying that the proposed technique is an effective way for developing simple finite element models with high performance.

New way to construct high order Hamiltonian variational integrators

This paper develops a new approach to construct variational integrators. A simplified unconventional Hamilton＇s variational principle corresponding to initial value problems is proposed, which is convenient for applications. The displacement and mo- mentum are approximated with the same Lagrange interpolation. After the numerical integration and variational operation, the original problems are expressed as algebraic equations with the displacement and momentum at the interpolation points as unknown variables. Some particular variational integrators are derived. An optimal scheme of choosing initial values for the Newton-Raphson method is presented for the nonlinear dynamic system. In addition, specific examples show that the proposed integrators are symplectic when the interpolation point coincides with the numerical integration point, and both are Gaussian quadrature points. Meanwhile, compared with the same order symplectic Runge-Kutta methods, although the accuracy of the two methods is almost the same, the proposed integrators are much simpler and less computationally expensive.

The Hamilton System and Hamilton Type Generalized Variational Principle for the Laminated Composite Plates and Shells

By introducing the Hamilton theory and algorithms into the problems of laminated composite plates andshells, the Hamiltion type generalized variational principle is established, and the Hamilton canonical equations andthe boundary conditions for the static and elastoplastic analysis of composite plates are presented. With thetransformation of phase variables, the Hamilton canonical equations and their boundary conditions for thecylindrical shells and doubly curved shells in the curvilinear coordinate are given.

Fractional Noether theorem and fractional Lagrange equation of multi-scale mechano-electrophysiological coupling model of neuron membrane

Noether theorem is applied to a variable order fractional multiscale mechano-electrophysiological model of neuron membrane dynamics.The variable orders fractional Lagrange equation of a multiscale mechano-electrophysiological model of neuron membrane dynamics is given.The variable orders fractional Noether symmetry criterion and Noether conserved quantities are given.The forms of variable orders fractional Noether conserved quantities corresponding to Noether symmetry generators solutions of the model under different conditions are discussed in detail,and it is found that the expressions of variable orders fractional Noether conserved quantities are closely dependent on the external nonconservative forces and material parameters of the neuron.

Action Origin of the Cosmos

In physical information theory elementary objects are represented as correlation structures with oscillator properties and characterized by action. The procedure makes it possible to describe the photons of positive and negative charges by positive and negative real action;gravitons are represented in equal amounts by positive and negative real, i.e., virtual action, and the components of the vacuum are characterized by deactivated virtual action. An analysis of the currents in the correlation structures of photons of static Maxwell fields with wave and particle properties, of the Maxwell vacuum and of the gravitons leads to a uniform three-dimensional representation of the structure of the action. Based on these results, a basic structure consisting of a system of oscillators is proposed, which describe the properties of charges and masses and interact with the photons of static Maxwell fields and with gravitons. All properties of the elemental components of nature can thus be traced back to a basic structure of action. It follows that nature can be derived from a uniform structure and this structure of action must therefore also be the basis of the origin of the cosmos.

Fractional differential equations of motion in terms of combined Riemann-Liouville derivatives

In this paper, we focus on studying the fractional variational principle and the differential equations of motion for a fractional mechanical system. A combined Riemann-Liouville fractional derivative operator is defined, and a fractional Hamilton principle under this definition is established. The fractional Lagrange equations and the fractional Hamilton canonical equations are derived from the fractional Hamilton principle. A number of special cases are given, showing the universality of our conclusions. At the end of the paper, an example is given to illustrate the application of the results.

RESTUDY OF COUPLED FIELD THEORIES FOR MICROPOLAR CONTINUA (Ⅱ)-THERMOPIEZOELECTRICITY AND MAGNETOTHERMOELASTICITY

The theories of thermopiezoelectricity and magnetoelasticity for micropolar continua have been systematically developed by W. Nowacki. In this paper, the theories are restudied. The reason why they were restricted to linear cases is analyzed. The more general conservation principle of energy, energy balance equation and Hamilton principle of thermopiezoelectricity and magnetoelasticity for micropolar continua are established. The corresponding complete equations of motion and boundary conditions as well as balance equations of energy rate for local and nonlocal micropolar thermopiezoelectricity and magnetothermoelasticity are naturally derived. By means of two new functionals and total variation the boundary conditions of displacement, microrotation, electric potential and temperature are also given.

RESTUDY OF COUPLED FIELD THEORIES FOR MICROPOLAR CONTINUA (Ⅰ)──MICROPOLAR THERMOELASTICITY

Problems of micropolar thermoelasticity have been presented and discussed by some authors in the traditional framework of micropolar continuum field theory. In this paper the theory of micropolar thermoelasticity is restudied. The reason why it was restricted to a linear one is analyzed. The rather general principle of virtual work and the new formulation for the virtual work of internal forces as well as the rather complete Hamilton principle in micropolar thermoelasticity are established. From this new Hamilton principle not only the equations of motion, the balance equation of entropy, the boundary conditions of stress, couple stress and heat, but also the boundary conditions of displacement, microrotation and temperature are simultaneously derived.

Effect of rotary inertia on stability of axially accelerating viscoelastic Rayleigh beams

The dynamic stability of axially moving viscoelastic Rayleigh beams is pre- sented. The governing equation and simple support boundary condition are derived with the extended Hamilton＇s principle. The viscoelastic material of the beams is described as the Kelvin constitutive relationship involving the total time derivative. The axial tension is considered to vary longitudinally. The natural frequencies and solvability condition are obtained in the multi-scale process. It is of interest to investigate the summation parametric resonance and principal parametric resonance by using the Routh-Hurwitz criterion to obtain the stability condition. Numerical examples show the effects of viscos- ity coefficients, mean speed, beam stiffness, and rotary inertia factor on the summation parametric resonance and principle parametric resonance. The differential quadrature method （DQM） is used to validate the value of the stability boundary in the principle parametric resonance for the first two modes.

NONLINEAR ANALYSIS ON THE VIBRATION OF ELASTIC PLATES

We consider the vibration of elastic thin plates under certain reasonable assumptions. We derive the nonlinear equations for this model by the Hamilton Principle. Under the conditions on the hyperbolicity for the initial data, we establish the local time wellposedness for the initial and boundary value problem by Picard iteration scheme, and obtain the estimates for the solutions.

Variation principle of piezothermoelastic bodies,canonical equation and homogeneous equation

Combining the symplectic variations theory, the homogeneous control equation and isopaxametric element homogeneous formulations for piezothermoelastic hybrid laminates problems were deduced. Firstly, based on the generalized Hamilton variation principle, the non-homogeneous Hamilton canonical equation for piezothermoelastic bodies was derived. Then the symplectic relationship of variations in the thermal equilibrium formulations and gradient equations was considered, and the non-homogeneous canonical equation was transformed to homogeneous control equation for solving independently the coupling problem of piezothermoelastic bodies by the incensement of dimensions of the canonical equation. For the convenience of deriving Hamilton isopaxametric element formulations with four nodes, one can consider the temperature gradient equation as constitutive relation and reconstruct new variation principle. The homogeneous equation simplifies greatly the solution programs which axe often performed to solve nonhomogeneous equation and second order differential equation on the thermal equilibrium and gradient relationship.

Analysis of nonlinear stability and post-buckling for Euler-type beam-column structure

Based on the assumption of finite deformation, the Hamilton variational principle is extended to a nonlinear elastic Euler-type beam-column structure located on a nonlinear elastic foundation. The corresponding three-dimensional （3D） mathematical model for analyzing the nonlinear mechanical behaviors of structures is established, in which the effects of the rotation inertia and the nonlinearity of material and geometry are considered. As an application, the nonlinear stability and the post-buckling for a linear elastic beam with the equal cross-section located on an elastic foundation are analyzed. One end of the beam is fully fixed, and the other end is partially fixed and subjected to an axial force. A new numerical technique is proposed to calculate the trivial solution, bifurcation points, and bifurcation solutions by the shooting method and the Newton- Raphson iterative method. The first and second bifurcation points and the corresponding bifurcation solutions are calculated successfully. The effects of the foundation resistances and the inertia moments on the bifurcation points are considered.

Dynamic equations for curved submerged floating tunnel

In virtue of reference Cartesian coordinates, geometrical relations of spatial curved structure are presented in orthogonal curvilinear coordinates. Dynamic equations for helical girder are derived by Hamilton principle. These equations indicate that four generalized displacements are coupled with each other. When spatial structure degenerates into planar curvilinear structure, two generalized displacements in two perpendicular planes are coupled with each other. Dynamic equations for arbitrary curvilinear structure may be obtained by the method used in this paper.

LAGRANGIAN MECHANICS ON KAHLER MANIFOLDS

Lagrangian mechanics on Kahler manifolds were discussed, and the complex mathematical aspects of Lagrangian operator, Lagrange＇s equation, the action functional, Hamilton＇ s principle, Hamilton＇ s equation and so on were given.

Intuitive Concept or Physical Meaning of Lagrangian

The authors have been studying on the principle of motion generation behind animals, mainly human, and have reached a certain milestone with it in [1]. Because [1] ended up being very interdisciplinary, the author has been looking for an opportunity to close in on the part where we have grasped the conceptual idea of a Lagrangian. This paper proposes the physical meaning or its intuitive concept of a Lagrangian. This is a daring attempt because the topic is over 240 years of enigma, whereby so many have neglected of its absence, and physics has gone further towards its frontiers of their time, and has successfully flourished. Meanwhile, Lagrangian is not getting enough of teachers’ attention on students getting stuck on this function, despite the fact that it is a strong foundation as is only the beginning towards Hamiltonian formalism, general relativity, and modern physics of today. This paper’s sole motive is to answer what the title says in detail, helping each and everyone who faces Lagrangian for their first time. The paper is positioned to be a supplement for [1]. This literature had three topics bound into one. Out of the three, this document focuses in the part of the intuitive meaning of Lagrangian, since the paper had contents related to multiple disciplines. The author finds it worthy to discuss this topic in an independent, more detailed manner.

Building Blocks of Nature

By means of a representation of the elementary objects by the Lagrange density and by the commutators of the communication relations, correlations can be formed using the Fourier transform, which under the conditions of the Hamilton principle, describes correlation structures of the elementary objects with oscillator properties. The correlation structures obtained in this way are characterized by physical information, the essential component of which is the action. The correlation structures describe the physical properties and their interactions under the sole condition of the Hamilton’s principle. The structure, the properties and the interactions of elementary objects can be led back in this way to a fundamental four dimensional structure, which is therefore in their different modifications the building block of nature. With the presented method, an alternative interpretation of elementary physical effects to quantum mechanics is obtained. This report provides an overview of the fundamentals and statements of physical information theory and its consequences for understanding the nature of elementary objects.

Nonlinear dynamic analysis of a rotor/bearing/seal system

In this study a new dynamic model of a rotor system is established based on the Hamilton principle and the finite element method (FEM). We analyze the dynamic behavior of the rotor system with the coupled effects of the nonlinear oil film force, the nonlinear seal force, and the mass eccentricity of the disk. The equations of the motion are solved effectively using the fourth order Runge-Kutta method in MATLAB. The dynamic behavior of the system is illustrated by bifurcation diagrams, largest Lyapunov exponents, phase trajectory diagrams, and Poincaré maps. The numerical results show that the rotational speed of the rotor, the pressure drop in the seal, the seal length, the seal clearance, and the mass eccentricity of the disk are the key parameters that significantly affect the dynamic characteristics of the rotor system. The motion of the rotor system exhibits complex types of periodic, quasi-periodic, double-periodic, multi-periodic, and chaotic vibrations. This analysis can be used to guide the design of seal parameters and to diagnose the vibration of rotor/bearing/seal systems.

Free Vibration Analysis of Nanocomposite Plates Reinforced by Graded Carbon Nanotubes Based on First-Order Shear Deformation Plate Theory

Based on the Mindlin’s first-order shear deformation plate theory this paper focuses on the free vibration behavior of functionally graded nanocomposite plates reinforced by aligned and straight single-walled carbon nanotubes(SWCNTs).The material properties of simply supported functionally graded carbon nanotube-reinforced(FGCNTR)plates are assumed to be graded in the thickness direction.The effective material properties at a point are estimated by either the Eshelby-Mori-Tanaka approach or the extended rule of mixture.Two types of symmetric carbon nanotubes(CNTs)volume fraction profiles are presented in this paper.The equations of motion and related boundary conditions are derived using the Hamilton’s principle.A semianalytical solution composed of generalized differential quadrature(GDQ)method,as an efficient and accurate numerical method,and series solution is adopted to solve the equations of motions.The primary contribution of the present work is to provide a comparative study of the natural frequencies obtained by extended rule of mixture and Eshelby-Mori-Tanaka method.The detailed parametric studies are carried out to study the influences various types of the CNTs volume fraction profiles,geometrical parameters and CNTs volume fraction on the free vibration characteristics of FGCNTR plates.The results reveal that the prediction methods of effective material properties have an insignificant influence of the variation of the frequency parameters with the plate aspect ratio and the CNTs volume fraction.