scholarly journals Penetration into potential barriers in several dimensions

1937 ◽  
Vol 163 (915) ◽  
pp. 606-610 ◽  
Keyword(s):  
Refractive Index ◽  
Jacobi Equation ◽  
One Dimension ◽  
Practical Application ◽  
One Dimensional ◽  
Potential Barriers ◽  
Separable Problems ◽  
Total Reflexion ◽  
Similar Approximation ◽  
Hamilton Jacobi Equation

It is well known that in regions in which the refractive index varies sufficiently slowly, Schrödinger’s equation can be very simply treated by using its connexion with Hamilton-Jacobi’s differential equation. It is also known that a similar approximation is possible in regions of slowly varying imaginary refractive index (total reflexion). For the latter case the method was developed in papers by Jeffreys (1924), Wentzel (1926), Brillouin (1926) and Kramers (1926). These papers discuss also the behaviour of the wave function in the neighbourhood of the limit between the regions of real and imaginary refractive index. But although the connexion with the Hamilton-Jacobi equation holds in any number of dimensions, this equation can be solved by elementary means only in one dimension (or for problems that can by separation be reduced to one dimension), and for this reason the practical application of the method has so far been limited to one-dimensional or separable problems. In the present paper we discuss the case of more than one dimension and show that certain very simple inequalities may be obtained.

The use of schlieren and shadowgraph techniques in the study of flow patterns in density stratified liquids

1967 ◽  
Vol 27 (3) ◽  
pp. 595-608 ◽  
Author(s):  
D. E. Mowbray
Keyword(s):  
Refractive Index ◽  
Quantitative Evaluation ◽  
Salt Solution ◽  
Flow Patterns ◽  
One Dimension ◽  
Lagrange Equations ◽  
One Dimensional ◽  
Schlieren System ◽  
Fluid Media ◽  
Dimensional Variation

Various methods of establishing fluid media stably stratified in either temperature or the concentration of a foreign constituent are compared. A method used to construct a salt solution with precisely predetermined stratification is described. The Euler-Lagrange equations are solved exactly for a medium whose refractive index varies linearly in one dimension; the solution shows that a schlieren system may be used. For media with non-linear one-dimensional variation of refractive index, a modified shadowgraph technique may be employed. A device for the quantitative evaluation of a one-dimensional stratification is outlined and an example of its use is given.

The Hamilton-Jacobi Equation Applied to Continuum

1997 ◽  
Vol 64 (3) ◽  
pp. 658-663 ◽  
Author(s):  
C. M. Leech
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Jacobi Equation ◽  
Principal Function ◽  
Partial Differential ◽  
One Dimensional ◽  
Linear Elastic ◽  
Elastic Bar ◽  
Continuum Systems ◽  
Hamilton Jacobi Equation

The Hamilton-Jacobi partial differential equation is established for continuum systems; to do this a new concept in material distributions is introduced. The Lagrangian and Hamiltonian are developed, so that the Hamilton-Jacobi equation can be formulated and the principal function defined. Finally the principal function is constructed for the dynamics of a one-dimensional linear elastic bar; the solution for its’ vibrations is then established following the differentiation of the principal function.

Finite Volume Hermite WENO Schemes for Solving the Hamilton-Jacobi Equation

2014 ◽  
Vol 15 (4) ◽  
pp. 959-980 ◽  
Author(s):  
Jun Zhu ◽  
Jianxian Qiu
Keyword(s):  
Finite Volume ◽  
Jacobi Equation ◽  
Two Dimensions ◽  
One Dimension ◽  
Weno Schemes ◽  
Numerical Tests ◽  
New Type ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

AbstractIn this paper, we present a new type of Hermite weighted essentially non-oscillatory (HWENO) schemes for solving the Hamilton-Jacobi equations on the finite volume framework. The cell averages of the function and its first one (in one dimension) or two (in two dimensions) derivative values are together evolved via time approaching and used in the reconstructions. And the major advantages of the new HWENO schemes are their compactness in the spacial field, purely on the finite volume framework and only one set of small stencils is used for different type of the polynomial reconstructions. Extensive numerical tests are performed to illustrate the capability of the methodologies.

scholarly journals The one-dimensional harmonic oscillator damped with Caldirola-Kanai Hamiltonian

2018 ◽  
Vol 64 (1) ◽  
pp. 47
Author(s):  
Francis Segovia-Chaves
Keyword(s):  
Characteristic Function ◽  
Harmonic Oscillator ◽  
Canonical Transformation ◽  
Exponential Growth ◽  
Jacobi Equation ◽  
One Dimensional ◽  
Harmonic Motion ◽  
Damped Harmonic Oscillator ◽  
The One ◽  
Hamilton Jacobi Equation

In this paper, the solution to the Hamilton-Jacobi equation for the one-dimensional harmonic oscillator damped with the Caldirola-Kanai model is presented. Making use of a canonical transformation, we calculate the Hamilton characteristic function. It was found that the position of the oscillator shows an exponential decay similar to that of the oscillator with damping where the decay is more pronounced when increasing the damping constant γ. It is shown that when γ = 0, the behavior is of an oscillator with simple harmonic motion. However, unlike the damped harmonic oscillator where the linear momentum decays with time, in the case of the oscillator with the Caldirola-KanaiHamiltonian, the momentum increases as time increases due to an exponential growth of the mass m(t) = meγt.

scholarly journals Existence of viscosity solutions with the optimal regularity of a two-peakon Hamilton–Jacobi equation

2021 ◽  
Vol 18 (02) ◽  
pp. 493-510
Author(s):  
Tomasz Cieślak ◽  
Jakub Siemianowski
Keyword(s):  
Jacobi Equation ◽  
Optimal Regularity ◽  
Hölder Continuous ◽  
Lipschitz Regularity ◽  
One Dimensional ◽  
Lipschitz Continuous ◽  
Dimension Formula ◽  
The One ◽  
The Value Function ◽  
Hamilton Jacobi Equation

We study here a Hamilton–Jacobi equation with a quadratic and degenerate Hamiltonian, which comes from the dynamics of a multipeakon in the Camassa–Holm equation. It is given by a quadratic form with a singular positive semi-definite matrix. We increase the regularity of the value function considered in earlier works, which is known to be the viscosity solution. We prove that for a two-peakon Hamiltonian such solutions are actually [Formula: see text]-Hölder continuous in space and time-Lipschitz continuous. The time-Lipschitz regularity is proven in any dimension [Formula: see text]. Such a regularity is already known in the one-dimensional case and, moreover it is the best possible, as shown earlier.

scholarly journals The Stäckel theorem in the Lagrangian formalism and the use of local times

2021 ◽  
Vol 67 (3 May-Jun) ◽  
pp. 447
Author(s):  
G. F. Torres del Castillo
Keyword(s):  
Jacobi Equation ◽  
Degrees Of Freedom ◽  
Local Times ◽  
Lagrangian Formalism ◽  
One Dimensional ◽  
Constants Of Motion ◽  
Hamilton Jacobi Equation

We show that the conditions for the separability of the Hamilton-Jacobi equation given by the Stäckel theorem imply that, making use of the elementary Lagrangian formalism, one can find $n$ functionally independent constants of motion, where $n$ is the number of degrees of freedom. We also show that this result can be linked to the fact that the Lagrangian for a system of this class is related to the sum of $n$ one-dimensional Lagrangians, if one makes use of multiple local times.

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