Hamilton's Canonical Equations Research Articles

Hamiltonian formulation of linear non-Hermitian systems

In the case of a linear non-Hermitian system, I prove that it's possible to construct a Hamiltonian in such a way that the equations governing the non-Hermitian system can be exactly expressed using Hamilton's canonical equations. Initially, I demonstrate this within the discrete representation framework and subsequently extend it to continuous representation. Through this formulation employing the Hamiltonian, I can pinpoint a conserved charge using Noether's theorem and identify adiabatic invariants. When this approach is applied to Hermitian systems, all the obtained results converge to the well-known outcomes associated with the Schrödinger equation.

Hamiltonian formulation of linear non-Hermitian systems

Abstract In the case of a linear non-Hermitian system, I prove that it's possible to construct a Hamiltonian in such a way that the equations governing the non-Hermitian system can be exactly expressed using Hamilton's canonical equations. Initially, I demonstrate this within the discrete representation framework and subsequently extend it to continuous representation. Through this formulation employing the Hamiltonian, I can pinpoint a conserved charge using Noether's theorem and identify adiabatic invariants. When this approach is applied to Hermitian systems, all the obtained results converge to the well-known outcomes associated with the Schr"odinger equation.

The Dynamical and Kinetic Equations of Four-Five-Six-Wave Resonance for Ocean Surface Gravity Waves in Water with a Finite Depth

Based on the Hamilton canonical equations for ocean surface waves with four-five-six-wave resonance conditions, the determinate dynamical equation of four-five-six-wave resonances for ocean surface gravity waves in water with a finite depth is established, thus leading to the elimination of the nonresonant second-, third-, fourth-, and fifth-order nonlinear terms though a suitable canonical transformation. The four kernels of the equation and the 18 coefficients of the transformation are expressed in explicit form in terms of the expansion coefficients of the gravity wave Hamiltonian in integral-power series in normal variables. The possibilities of the existence of integrals of motion for the wave momentum and the wave action are discussed, particularly the special integrals for the latter. For ocean surface capillary–gravity waves on a fluid with a finite depth, the sixth-order expansion coefficients of the Hamiltonian in integral-power series in normal variables are concretely provided, thus naturally including the classical fifth-order kinetic energy expansion coefficients given by Krasitskii.

Symplectic superposition method for the free-vibrating problem of sigmoid functionally graded material rectangular thin plates clamped at four edges

The symplectic superposition method (SSM) is implemented to investigate the free vibration problem of sigmoid functionally graded material (FGM) rectangular thin plates clamped at four edges. Firstly, the vibration equation of the sigmoid FGM rectangular thin plate is transformed into Hamiltonian canonical equations. Then based on the analysis of boundary conditions of the plate, the vibration problem of the sigmoid FGM rectangular thin plate clamped at four edges is decomposed as two sub-problems simply supported on two opposite edges. And then based on the symplectic elasticity method (SEM), the series expansion solutions for the two sub-vibrating problems are obtained. Afterward, by superposing the series expansion solutions for the two sub-vibrating problems, a symplectic superposition solution of the freely vibrating sigmoid FGM rectangular thin plates clamped at four edges is obtained. Finally, we apply the obtained symplectic superposition solution to research the frequencies and vibration mode functions of some specific sigmoid FGM rectangular thin plates, and the impact of the aspect ratio a/b and material gradient index k on the vibration frequencies of the sigmoid FGM rectangular thin plates is studied.

A canonical Hamiltonian formulation of the Navier–Stokes problem

This paper presents a novel Hamiltonian formulation of the isotropic Navier–Stokes problem based on a minimum-action principle derived from the principle of least squares. This formulation uses the velocities $u_{i}(x_{j},t)$ and pressure $p(x_{j},t)$ as the field quantities to be varied, along with canonically conjugate momenta deduced from the analysis. From these, a conserved Hamiltonian functional $H^{*}$ satisfying Hamilton's canonical equations is constructed, and the associated Hamilton–Jacobi equation is formulated for both compressible and incompressible flows. This Hamilton–Jacobi equation reduces the problem of finding four separate field quantities ( $u_{i}$ , $p$ ) to that of finding a single scalar functional in those fields – Hamilton's principal functional ${S}^{*}[u_{i},p,t]$ . Moreover, the transformation theory of Hamilton and Jacobi now provides a prescribed recipe for solving the Navier–Stokes problem: find ${S}^{*}$ . If an analytical expression for ${S}^{*}$ can be obtained, it will lead via canonical transformation to a new set of fields which are simply equal to their initial values, giving analytical expressions for the original velocity and pressure fields. Failing that, if one can only show that a complete solution to this Hamilton–Jacobi equation does or does not exist, that will also resolve the question of existence of solutions. The method employed here is not specific to the Navier–Stokes problem or even to classical mechanics, and can be applied to any traditionally non-Hamiltonian problem.

Symplectic solutions of the plane annular sectors in micropolar elasticity

The symplectic approach is utilized to derive the solutions of the plane annular sector in micropolar elasticity. According to the Hellinger-Reissner variational principle, the Hamiltonian canonical equations are obtained for the plane problem in an annular sectorial region. Through the method of separation of variables, we obtain an eigenvalue equation of the Hamiltonian operator matrix which is a system of linear differential equations with variable coefficients. By the method of series solutions, its general solutions can be expressed as the linear combination of Bessel functions and power functions. The corresponding eigensolutions for three types of homogeneous boundary conditions are derived. According to the adjoint symplectic orthogonality of the eigensolutions, the solution of the micropolar plane annular sector is expressed as a series expansion of eigensolutions. Numerical analyses are presented to demonstrate the validity of the proposed method and confirm its ability to accurately capture the size effect.

Analytical solution for the bending problem of micropolar plates based on the symplectic approach

Abstract An analytical solution for the bending problem of micropolar plates is derived based on the symplectic approach. By applying Legendre's transformation, we obtain the Hamiltonian canonical equation for the bending problem of a micropolar plate. According to the method of separation of variables, the homogenous Hamiltonian canonical equation can be transformed into an eigenvalue problem of the Hamiltonian operator matrix. We derive the eigensolutions of the eigenvalue problem for the simply-supported, free and clamped boundary conditions at the two opposite sides. Based on the adjoint symplectic orthogonal relation of the eigensolutions, the solution of the bending problem of micropolar plate is expressed as a series expansion of eigensolutions. Numerical results confirm the validity of the present approach for the bending problem of micropolar plates under various boundary conditions and demonstrate the capability of the proposed approach to capture the size-dependent behavior of micropolar plates.

Liouville equation in statistical mechanics is not applicable to gases composed of colliding molecules

Liouville equation is a fundamental one in statistical mechanics. It is rooted in ensemble theory. By ensemble theory, the variation of the system’s microscopic state is indicated by the moving of the phase point, and the moving trajectory is believed continuous. Thus, the ensemble density is thought to be a smooth function, and it observes continuity equation. When the Hamiltonian canonical equations of the molecules are applied to the continuity equation, Liouville equation can be obtained. We carefully analyze a gas composed of a great number of molecules colliding with each other. The defects in deriving Liouville equation are found. Due to collision, molecules’ momenta changes discontinuously, so that the trajectories of the phase points are actually not continuous. In statistical mechanics, infinitesimals in physics and in mathematics should be distinguished. In continuity equation that the ensemble density satisfies, the derivatives with respect to space and time should be physical infinitesimals, while in Hamiltonian canonical equations that every molecule follows, the derivatives take infinitesimals in mathematics. In the course of deriving Liouville equation, the infinitesimals in physics are unknowingly replaced by those in mathematics. The conclusion is that Liouville equation is not applicable to gases.

The frozen orbits of the charged satellites under zonal harmonics perturbation

In the present work, the problem of frozen orbits of a charged satellite around the Earth is treated. The geopotential is considered as oblate body up to J6 zonal harmonics. The first normalized Hamiltonian is utilized so as to compute families of the long-term frozen orbits for a charged satellite in the concerned model using the Lie transform method. Since the mathematical model of the problem is complicated, the Hamilton canonical equations have been solved numerically. Our numerical investigation of the considered dynamical system reveals no frozen orbits can be obtained for orbits with perigee point lies on the equator g∈0.0°, 0.180°. But, in the neighbouhood of these orbits, equilibria for a very nearly circular orbits as well as equatorial HEO exist. Our results reflected the remarkable effects of the electromagnetic perturbations on the frozen orbit dynamics, especially for the medium and high Earth orbits. Investigating charged satellites carrying different electronic charges, the phase space (I,e) is noticeably changed near the equilibria g=n(180°),n=0,1. It is observed also that a very little change in the amount of charge carried by the satellite, gives rise to a change in the phase space especially for the nearly circular orbits that have inclinations in between the two roots of critical of inclinations.

Research on elastodynamic theory based on the framework of energy conservation

Based on the analysis of establishing dynamic equations by using Newton's mechanics, Lagrange's, and Hamilton's mechanics, a new idea of establishing elastodynamic equations under the framework of energy conservation is proposed. Firstly, Newton’s second law is used to derive wave equations of motion. Secondly, Lagrange's equation, Hamilton's canonical equations, and the corresponding dynamical equations in a continuum medium are derived by using Hamilton’s variational principle. Thirdly, under the framework of energy conservation, Lagrange's equation, Hamilton's canonical equations, and the acoustic dynamic equations of the continuum are established, and the results are proved to be consistent with those derived from classical mechanics. Some fuzzy understandings when using Hamilton's variational principle to establish Lagrange’s equation and Hamilton’s canonical equation, are clarified. A series of dynamical equations established under the framework of energy conservation provides an alternative way to characterize and represent the propagation characteristics of wave motions in various complex media without involving the variational principle of functional extremum. Finally, as an application example, the differential equation of elastodynamics in a viscoelastic medium is given under the framework of energy conservation.

Optimal control of TGF-β to prevent formation of pulmonary fibrosis.

In this paper, three optimal control problems are proposed to prevent forming lung fibrosis while control is transforming growth factor-β (TGF-β) in the myofibroblast diffusion process. Two diffusion equations for fibroblast and myofibroblast are mathematically formulated as the system's dynamic, while different optimal control model problems are proposed to find the optimal TGF-β. During solving the first optimal control problem with the regulator objection function, it is understood that the control function gets unexpected negative values. Thus, in the second optimal control problem, for the control function, the non-negative constraint is imposed. This problem is solved successfully using the extended canonical Hamiltonian equations with no flux boundary conditions. Pontryagin's minimum principle is used to solve the related optimal control problems successfully. In the third optimal control problem, the fibroblast equation is added to a dynamic system consisting of the partial differential equation. The two-dimensional diffusion equations for fibroblast and myofibroblast are transferred to a system of ordinary differential equations using the central finite differences explicit method. Three theorems and two propositions are proved using extended Pontryagin's minimum principle and the extended Hamiltonian equations. Numerical results are given. We believe that this optimal strategy can help practitioners apply some medication to reduce the TGF-β in preventing the formation of pulmonary fibrosis.

The symplectic approach for analytical solution of micropolar plane stress problem

The symplectic method is adopted to derive analytical solution of the plane stress problem in the framework of the micropolar elasticity. The general micropolar plane stress problem is transformed into three types of problems with homogeneous boundary conditions at two opposite edges. The fundamental equations are converted into canonical Hamiltonian equations using the Legendre transformation. The eigensolutions of the Hamiltonian operator matrix are obtained for three types of homogeneous boundary conditions. The analytical solution for the micropolar plane stress problem is expressed as an eigensolution expansion. The size effect is analyzed based on the relationship between the eigenvalues and micropolar material constants. The numerical results obtained by the finite element method (FEM) and ABAQUS demonstrate the validity of the proposed method for different boundary conditions and confirm its ability to accurately capture the size effect.

An Analytical Symplectic Method for Buckling of Ring-Stiffened Graphene Platelet-Reinforced Composite Cylindrical Shells Subjected to Hydrostatic Pressure

In this paper, a novel analytical approach for the buckling of ring-stiffened porous graphene platelet-reinforced composite cylindrical shells under hydrostatic pressure is proposed under the framework of symplectic mechanics. Three types of graphene platelet-reinforced patterns and porosity distributions are considered, and the effective material properties of porous graphene platelet-reinforced composite are determined with a modified Halpin–Tsai model. In the symplectic approach, the governing equations in the conventional Lagrangian system are transformed into a set of Hamiltonian canonical equations, and therefore, the buckling analysis is reduced into an eigenproblem in a symplectic space. Consequently, the accurate critical pressures and corresponding analytical buckling mode shapes are obtained simultaneously without any trial function. The numerical results are compared with the existing results, and good agreements are observed. A comprehensive parametric study of the geometrical parameters, boundary conditions, material properties, and ring-stiffener parameters on the buckling behavior of such shells is also presented.

Splitting Methods for Semi-Classical Hamiltonian Dynamics of Charge Transfer in Nonlinear Lattices

We propose two classes of symplecticity-preserving symmetric splitting methods for semi-classical Hamiltonian dynamics of charge transfer by intrinsic localized modes in nonlinear crystal lattice models. We consider, without loss of generality, one-dimensional crystal lattice models described by classical Hamiltonian dynamics, whereas the charge (electron or hole) is modeled as a quantum particle within the tight-binding approximation. Canonical Hamiltonian equations for coupled lattice-charge dynamics are derived, and a linear analysis of linearized equations with the derivation of the dispersion relations is performed. Structure-preserving splitting methods are constructed by splitting the total Hamiltonian into the sum of Hamiltonians, for which the individual dynamics can be solved exactly. Symmetric methods are obtained with the Strang splitting of exact, symplectic flow maps leading to explicit second-order numerical integrators. Splitting methods that are symplectic and conserve exactly the charge probability are also proposed. Conveniently, they require only one solution of a linear system of equations per time step. The developed methods are computationally efficient and preserve the structure; therefore, they provide new means for qualitative numerical analysis and long-time simulations for charge transfer by nonlinear lattice excitations. The properties of the developed methods are explored and demonstrated numerically considering charge transport by mobile discrete breathers in an example model previously proposed for a layered crystal.

Symplectic Method for the Thin Piezoelectric Plates

The symplectic method for a thin piezoelectric plate problem is developed. The Hamiltonian canonical equation of thin piezoelectric plate is given by using the variational principle. By applying the separation of variables method, we can obtain symplectic orthogonal eigensolutions. As an application, the problem of a thin piezoelectric plate with full edges simply supported under a uniformly distributed load is discussed, and analytical solutions of the deflection and potential of a piezoelectric thin plate are obtained. A numerical example shows that the solutions converge very rapidly. The advantage of this method is that it does not need to assume the predetermined function in advance, so it has better universality. It may also be applied to the problem of thin piezoelectric plate buckling and vibrating.

Research on dynamics theory based on the framework of energy conservation

Based on the analysis of establishing dynamic equations by using Newton mechanics, Lagrange and Hamilton mechanics, a new idea of establishing dynamic equations under the framework of energy conservation is proposed. Firstly, the use of Newton’s second law to derive wave equations of motion is introduced. Secondly, the Lagrange equation, Hamilton canonical equation, and the corresponding dynamical equations in a continuum are derived by using Hamilton’s variational principle. Thirdly, under the framework of energy conservation, the Lagrange equation, Hamilton canonical equation, and acoustic dynamics equation of continuum are established, and the results are proved to be consistent with those derived from classical mechanics. Some of fuzzy understandings when using Hamilton's variational principle to establish Lagrange’s equation and Hamilton’s canonical equation, are clarified. A series of dynamical equations established under the framework of energy conservation provides alternative way to characterize and represent the propagation characteristics of wave motions in various complex media without involving the variational principle of functional extremum. Finally, as an application example, the differential equation of dynamics in viscoelastic medium is given under the framework of energy conservation.

Research on Shear Lag Warping Displacement Modes of Frame-Tube Structures Based on the Hamiltonian Mechanics

Based on the equivalent continuity method, the accuracy of the shear lag warping displacement functions for frame-tube structures was studied under the Hamiltonian mechanics. Different types of functions were selected to describe the shear lag warping displacement of the flange plate, and the shear deformation and longitudinal warping of the equivalent plate were considered. The total potential energy of the structure and the corresponding Lagrangian function under different displacement modes were obtained. Not with the traditional variational methods, the problem was studied under the Hamiltonian mechanics system. The Hamiltonian canonical equation for the frame-tube structure was derived and solved with the precise integration method, then the column axial force was calculated and the accuracy was analyzed. The verification results of the calculation examples show that, this method is simple and feasible to analyze the shear lag effects of the frame-tube structures. The choice of different warping displacement functions has little effect on the lateral displacement calculation results, but has great influence on the axial force solution, and the quadratic parabola can best reflect the actual warping displacement distribution of the flange. Comparison of the stress distributions in the equivalent flange under different types of loads indicate that, with the increase of the position of the external load resultant force, the negative shear-lag effect on the top gradually weakens to disappear.

Newton's second law in field theory

In this article we present a natural generalization of Newton's Second Law valid in field theory, i.e., when the parameterized curves are replaced by parameterized submanifolds of higher dimension. For it we introduce what we have called the geodesic k-vector field, analogous to the ordinary geodesic field and which describes the inertial motions (i.e., evolution in the absence of forces). From this generalized Newton's law, the corresponding Hamilton's canonical equations of field theory (Hamilton-De Donder-Weyl equations) are obtained by a simple procedure. It is shown that solutions of generalized Newton's equation also hold the canonical equations. However, unlike the ordinary case, Newton equations determined by different forces can define equal Hamilton's equations.

Figure-eight orbits in three post-Newtonian formulations of triple black holes

There are three types of dynamical equations for a post-Newtonian Lagrangian formulation of three black holes. They are approximate post-Newtonian Lagrangian equations of motion, coherent post-Newtonian Lagrangian equations of motion, and post-Newtonian Hamiltonian canonical equations. The energy integral is conserved approximately in the first model, but exactly in the second and third models. The three formulations are not exactly equivalent, while they are approximately related in the orbital dynamical behavior. Figure-eight orbits in the three problems may have different initial conditions. Imai, Chiba, and Asada adjusted the initial velocities satisfying the figure-eight orbits for given initial positions in the first model. The initial velocities are still suitable for the figure-eight orbits of the latter two models when the initial separations of the first body are enough large. However, they are not generally applicable to relatively small initial separations in the three systems. We can obtain the desired initial velocities by scanning the initial velocities of the third body in the three systems. The figure-eight orbits in these models exhibit sensitive dependence on the scanned initial velocities for smaller initial separations. Although the differences in the scanned initial velocities among the three problems are minor for the small initial separations, the scanned initial velocities for any one of the three systems do not yield the figure-eight orbits for the other two systems.

Large Deformation Dynamic Analysis of a Cable System by a New Hamiltonian Finite Element Method

This paper develops a new Hamiltonian nodal position finite element method for dynamic analysis of spatial flexible cable systems with large deformation. The dynamic governing equation is derived from finite elasticity theory. Logarithmic strain is applied to construct large deformation Hamiltonian canonical equations. An efficient second-order symplectic difference algorithm is built to solve the canonical equations numerically. A large strain conical pendulum system is analyzed numerically by the proposed method, and the numerical results are compared with those retracted from the existing Hamiltonian methods and Livermore Software Technology Corporation: dynamics (LS-DYNA). The proposed method is further verified by two tethered dynamic experiments involving large displacement motion and large deformation. The comparisons and verifications demonstrate that the proposed method is of symplectic conservation, has high accuracy and has stability for calculating flexible cable system dynamics with large deformation.