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.

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 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.

scholarly journals The Hamilton-Jacobi equation and holographic renormalization group flows on sphere

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Nakwoo Kim ◽  
Se-Jin Kim
Keyword(s):  
Renormalization Group ◽  
Characteristic Function ◽  
Jacobi Equation ◽  
Degrees Of Freedom ◽  
Warp Factor ◽  
Holographic Renormalization ◽  
Mass Terms ◽  
Renormalization Group Flows ◽  
Jacobi Formulation ◽  
Hamilton Jacobi Equation

Abstract We study the Hamilton-Jacobi formulation of effective mechanical actions associated with holographic renormalization group flows when the field theory is put on the sphere and mass terms are turned on. Although the system is supersymmetric and it is described by a superpotential, Hamilton’s characteristic function is not readily given by the superpotential when the boundary of AdS is curved. We propose a method to construct the solution as a series expansion in scalar field degrees of freedom. The coefficients are functions of the warp factor to be determined by a differential equation one obtains when the ansatz is substituted into the Hamilton-Jacobi equation. We also show how the solution can be derived from the BPS equations without having to solve differential equations. The characteristic function readily provides information on holographic counterterms which cancel divergences of the on-shell action near the boundary of AdS.

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.

Center of Mass Theorem and the Separation of Variables for Two-Body System on a Three-Dimensional Sphere

2020 ◽  
Vol 23 (3) ◽  
pp. 306-311
Author(s):  
Yu. Kurochkin ◽  
Dz. Shoukavy ◽  
I. Boyarina
Keyword(s):  
Constant Curvature ◽  
Jacobi Equation ◽  
Separation Of Variables ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Relative Distance ◽  
Dimensional Sphere ◽  
Body System ◽  
Spaces Of Constant Curvature ◽  
Hamilton Jacobi Equation

The immobility of the center of mass in spaces of constant curvature is postulated based on its definition obtained in [1]. The system of two particles which interact through a potential depending only on the distance between particles on a three-dimensional sphere is considered. The Hamilton-Jacobi equation is formulated and its solutions and trajectory equations are found. It was established that the reduced mass of the system depends on the relative distance.

Sign in / Sign up

Export Citation Format

Share Document