Tunneling theory without the transfer-Hamiltonian formalism. IV. The abrupt (zero-width) three-dimensional junction

1976 ◽  
Vol 13 (2) ◽  
pp. 517-530 ◽  
Author(s):  
T. E. Feuchtwang
Keyword(s):  
Three Dimensional ◽  
Hamiltonian Formalism ◽  
Tunneling Theory

scholarly journals Spatially homogeneous black hole solutions in $$z=4$$ Hořava–Lifshitz gravity in $$(4+1)$$ dimensions with Nil geometry and $$H^2\times R$$ horizons

2021 ◽  
Vol 81 (10) ◽  
Author(s):  
F. Naderi ◽  
A. Rezaei-Aghdam ◽  
Z. Mahvelati-Shamsabadi
Keyword(s):  
Black Hole ◽  
Black Hole Solution ◽  
Three Dimensional ◽  
Hamiltonian Formalism ◽  
Coupling Constant ◽  
Especial Case ◽  
Gravity Equations ◽  
Spatially Homogeneous ◽  
Black Hole Solutions ◽  
Bianchi Types

AbstractIn this paper, we present two new families of spatially homogeneous black hole solution for $$z=4$$ z = 4 Hořava–Lifshitz Gravity equations in $$(4+1)$$ ( 4 + 1 ) dimensions with general coupling constant $$\lambda $$ λ and the especial case $$\lambda =1$$ λ = 1 , considering $$\beta =-1/3$$ β = - 1 / 3 . The three-dimensional horizons are considered to have Bianchi types II and III symmetries, and hence the horizons are modeled on two types of Thurston 3-geometries, namely the Nil geometry and $$H^2\times R$$ H 2 × R . Being foliated by compact 3-manifolds, the horizons are neither spherical, hyperbolic, nor toroidal, and therefore are not of the previously studied topological black hole solutions in Hořava–Lifshitz gravity. Using the Hamiltonian formalism, we establish the conventional thermodynamics of the solutions defining the mass and entropy of the black hole solutions for several classes of solutions. It turned out that for both horizon geometries the area term in the entropy receives two non-logarithmic negative corrections proportional to Hořava–Lifshitz parameters. Also, we show that choosing some proper set of parameters the solutions can exhibit locally stable or unstable behavior.

Advanced Plate Theory for Multibody Dynamics

2013 ◽  
Author(s):  
Olivier A. Bauchau ◽  
Shilei Han
Keyword(s):  
Plate Theory ◽  
Three Dimensional ◽  
Hamiltonian Formalism ◽  
Laminated Composite ◽  
Composite Plates ◽  
Complex Stress State ◽  
Mindlin Plate ◽  
Element Discretization ◽  
Deformation State ◽  
Material Line

In flexible multibody systems, many components are often approximated as plates. More often that not, classical plate theories, such as Kirchhoff or Reissner-Mindlin plate theory, form the basis of the analytical development for plate dynamics. The advantage of this approach is that it leads to a very simple kinematic representation of the problem: the plate’s normal material line is assumed to remain straight and its displacement field is fully defined by three displacement and two rotation components. While such approach is capable of capturing the kinetic energy of the system accurately, it cannot represent the strain energy adequately. For instance, it is well known from three-dimensional elasticity theory that the normal material line will warp under load for laminated composite plates, leading to a three-dimensional deformation state that generates a complex stress state. To overcome this problem, several high-order and refined plate theory were proposed. While these approaches work well for some cases, they typically lead to inefficient formulation because they introduce numerous additional variables. This paper presents a different approach to the problem, which is based on a finite element discretization of the normal material line, and relies of the Hamiltonian formalism of obtain solutions of the governing equations. Polynomial solutions, also known as central solutions, are obtained that propagate over the entire span of the plate.

A dynamic equation for 2D surface waves on deep water

2020 ◽  
Author(s):  
Dmitry Kachulin ◽  
Alexander Dyachenko ◽  
Vladimir Zakharov
Keyword(s):  
Fourth Order ◽  
Canonical Transformation ◽  
Three Dimensional ◽  
Fourier Method ◽  
Hamiltonian Formalism ◽  
Two Dimensional ◽  
Russian Science ◽  
Wave Interactions ◽  
Zakharov Equation ◽  
Limiting Case

<p>Using the Hamiltonian formalism and the theory of canonical transformations, we have constructed a model of the dynamics of two-dimensional waves on the surface of a three-dimensional fluid. We find and apply a canonical transformation to a water wave equation to remove all nonresonant cubic and fourth-order nonlinear terms. The found canonical transformation also allows us to significantly simplify the fourth-order terms in the Hamiltonian by replacing the coefficient of four-wave Zakharov interactions with a new simpler one. As a result, unlike the Zakharov equation (written in k-space), this equation can be written in x-space, which greatly simplifies its numerical simulation. In addition, our chosen form of a new coefficient of four-wave interactions allows us to generalize this equation to describe two-dimensional waves on the surface of a three-dimensional fluid. An effective numerical algorithm based on the pseudospectral Fourier method for solving the new 2D equation is developed. In the limiting case of plane (one-dimensional) waves, we found solutions in the form of breathers propagating in one direction. The dynamics of such nonlinear traveling waves perturbed in the transverse direction is numerically investigated.</p><p>The work was supported by the Russian Science Foundation (Grant No. 19-72-30028).</p>

scholarly journals Partial-mixed special finite element for the analysis of multilayer composites and FGM

2012 ◽  
Author(s):  
Orlando Andrianarison ◽  
Ayech Benjeddou
Keyword(s):  
Finite Element ◽  
Three Dimensional ◽  
Hamiltonian Formalism ◽  
Single Layer ◽  
Static Problem ◽  
Functionally Graded ◽  
Elasticity Equations ◽  
Graded Material ◽  
Transverse Stresses ◽  
Three Dimensional Elasticity

A partial-mixed special finite element (FE) is proposed for the static analysis of multilayer composite and functionally graded material plates. Using the Hamiltonian formalism, the three-dimensional elasticity equations are first reformulated so that a partial-mixed variational formulation, retaining as primary variables the translational displacements augmented with the transverse stresses only, is obtained; this allows, in particular, a straightforward fulfilment of the multilayer interfaces continuity conditions. After an only in-plane FE discretisation, the static problem is then reduced, for a single layer, to a Hamiltonian eigenvalue problem that is solved analytically, through the layer thickness, using the symplectic formalism; the multilayer solution is finally reached via the state-space method and the propagator matrix concept. The performance, in convergence and accuracy, of the proposed approach, applied to representative examples, is shown to be very satisfactory.

scholarly journals LONGITUDINAL TRAP OF ELECTRON BEAM IN POTENTIAL PIT MAGNETIC SOLENOIDAL FIELD

2021 ◽  
pp. 62-66
Author(s):  
Oleksandr Mazmanishvili ◽  
Nikolay Reshetnyak
Keyword(s):  
Magnetic Field ◽  
Numerical Simulation ◽  
Electron Beam ◽  
Electron Flow ◽  
Three Dimensional ◽  
Hamiltonian Formalism ◽  
Polar Angle ◽  
Longitudinal Motion ◽  
Gradient Magnetic Field ◽  
The Magnetic Field

A two-mode cylindrical magnetic field is considered, the potential of which has a minimum. The object of this work is the study of the parameters of an electron beam when it moves in a solenoid field with the longitudinal trap formed by the magnetic field, and the construction of the computational model of the motion of an electron beam. The problem is posed of the stability of the motion of electrons in such solenoid magnetic field. The possibility of obtaining oscillatory modes of particle motion has been studied. It was found that for oscillations of particles with an energy of tens of kiloelectronvolts in the potential well in a well, the field with the amplitude of tens of thousands of Oersteds is required. For the solenoid magnetic field of the solenoid, the formation of electron beam with an energy of 55 keV in the longitudinal and radial directions during its transportation is studied. A section of a magnetron gun was used as the physical object. One possible direction is to combine the two matched magnetic systems of the gun to create the potential magnetic field well. It is shown that, for the chosen conditions, the motion of electrons can be associated with the model of three-dimensional oscillations. In this work, on the basis of the Hamiltonian formalism of the motion of electrons in a magnetic field and an algorithm for numerically finding solutions to the differential equations of dynamics, a software tool is constructed that allows one to obtain arrays of values of particle trajectories in the volume. The use of the software made it possible to simulate the main dependences of the motion of the electron beam in a given two-mode solenoid magnetic field. The results of numerical simulation of electron trajectories in the gradient magnetic field with the point secondary emission cathode located in the middle of the system are presented. The formation of the beam with energy of 55 keV in the radial and longitudinal directions during its transportation in a solenoid magnetic field with a large gradient is considered. For significant time intervals, the possibility of three-dimensional oscillations is shown and the operating modes of the magnetic system are obtained, in which the particle undergoes stable three-dimensional oscillations. The influence of the initial conditions during emission on the occurrence of the reciprocating oscillatory effect has been studied. It is shown that for a given electron energy and fixed magnetic field, the parameter that determines the reflection of a particle, is the polar angle of entry relative to the axis of the cylindrical magnetic field. The dependence of the formation of the final distribution of particles on the amplitude and gradient of the magnetic field along the axis of the system is investigated. The results of numerical simulation on the motion of the electron flow are presented. The characteristics of the resulting electron beam are considered on the basis of a model of electron flow motion. The obtained simulation results show that it is possible to establish the phenomenon of oscillatory-return longitudinal motion under experimental conditions. Keywords: electron beam, magnetron gun, three-dimensional oscillations, electron dynamics, gradient magnetic field, mathematical modeling.

Hamiltonian Systems Based on a Lie Algebra

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Lie Algebra ◽  
Euler Equations ◽  
Burgers Equation ◽  
Kdv Equation ◽  
Three Dimensional ◽  
Hamiltonian Formalism ◽  
Virasoro Algebra ◽  
Inner Product ◽  
The Kdv Equation ◽  
Limiting Case

There is a remarkable analogy between Euler’s equations for a rigid body and his equations for an ideal fluid. The unifying idea is that of a Lie algebra with an inner product, which is not invariant, on it. The concepts of a vector space, Lie algebra, and inner product are reviewed. A hamiltonian dynamical system is derived from each metric Lie algebra. The Virasoro algebra (famous in string theory) is shown to lead to the KdV equation; and in a limiting case, to the Burgers equation for shocks. A hamiltonian formalism for two-dimensional Euler equations is then developed in detail. A discretization of these equations (using a spectral method) is then developed using mathematical ideas from quantum mechanics. Then a hamiltonian formalism for the full three-dimensional Euler equations is developed. The Clebsch variables which provide canonical pairs for fluid dynamics are then explained, in analogy to angular momentum.

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