Vector fields, line integrals, and the Hamilton-Jacobi equation: Semiclassical quantization of bound states

1984 ◽  
Vol 30 (1) ◽  
pp. 5-18 ◽  
Author(s):  
N. de Leon ◽  
E. J. Heller
Keyword(s):  
Jacobi Equation ◽  
Bound States ◽  
Vector Fields ◽  
Semiclassical Quantization ◽  
Hamilton Jacobi Equation

scholarly journals The Hamilton–Jacobi Theory for Contact Hamiltonian Systems

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1993
Author(s):  
Manuel de León ◽  
Manuel Lainz ◽  
Álvaro Muñiz-Brea
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Jacobi Equation ◽  
Vector Fields ◽  
Hamiltonian Function ◽  
Hamilton Jacobi Equation

The aim of this paper is to develop a Hamilton–Jacobi theory for contact Hamiltonian systems. We find several forms for a suitable Hamilton–Jacobi equation accordingly to the Hamiltonian and the evolution vector fields for a given Hamiltonian function. We also analyze the corresponding formulation on the symplectification of the contact Hamiltonian system, and establish the relations between these two approaches. In the last section, some examples are discussed.

Semiclassical quantization of non-Hermitian multidimensional systems using Hamilton–Jacobi equation

2007 ◽  
Vol 85 (12) ◽  
pp. 1481-1490 ◽  
Author(s):  
A Nanayakkara
Keyword(s):  
Jacobi Equation ◽  
Good Accuracy ◽  
Equation Of Motion ◽  
Multidimensional Systems ◽  
Semiclassical Quantization ◽  
Classical Trajectories ◽  
Action Variables ◽  
Hamilton Jacobi Equation

The Hamilton–Jacobi equation of motion is solved in action variables for non-Hermitian systems. Both real and complex semiclassical eigenvalues are obtained that make action variables into integers. This study shows, regardless of the existence of periodic or quasi-periodic classical trajectories, Hamilton–Jacobi methods can be applied to quantize some complex non-Hermitian systems with a good accuracy. PACS Nos.: 23.23.+x, 56.65.Dy

scholarly journals Structural aspects of Hamilton–Jacobi theory

2016 ◽  
Vol 13 (02) ◽  
pp. 1650017 ◽  
Author(s):  
J. F. Cariñena ◽  
X. Gràcia ◽  
G. Marmo ◽  
E. Martínez ◽  
M. C. Muñoz-Lecanda ◽  
...  
Keyword(s):  
Jacobi Equation ◽  
Vector Fields ◽  
Hamiltonian Dynamics ◽  
Dimensional Manifold ◽  
Special Cases ◽  
Fibered Manifolds ◽  
Hamilton Jacobi Equation ◽  
Structural Aspects ◽  
Lower Dimensional ◽  
Nonholonomic Dynamical Systems

In our previous papers [J. F. Cariñena, X. Gràcia, G. Marmo, E. Martínez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometric Hamilton–Jacobi theory, Int. J. Geom. Meth. Mod. Phys. 3 (2006) 1417–1458; Geometric Hamilton–Jacobi theory for nonholonomic dynamical systems, Int. J. Geom. Meth. Mod. Phys. 7 (2010) 431–454] we showed that the Hamilton–Jacobi problem can be regarded as a way to describe a given dynamics on a phase space manifold in terms of a family of dynamics on a lower-dimensional manifold. We also showed how constants of the motion help to solve the Hamilton–Jacobi equation. Here we want to delve into this interpretation by considering the most general case: a dynamical system on a manifold that is described in terms of a family of dynamics (slicing vector fields) on lower-dimensional manifolds. We identify the relevant geometric structures that lead from this decomposition of the dynamics to the classical Hamilton–Jacobi theory, by considering special cases like fibered manifolds and Hamiltonian dynamics, in the symplectic framework and the Poisson one. We also show how a set of functions on a tangent bundle can determine a second-order dynamics for which they are constants of the motion.

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.

Hamiltonian Mechanics

2017 ◽  
Author(s):  
Jennifer Coopersmith
Keyword(s):  
Angular Momentum ◽  
Jacobi Equation ◽  
Modern Theory ◽  
Hamiltonian Mechanics ◽  
Lagrangian Mechanics ◽  
Canonical Transformations ◽  
Canonical Variables ◽  
Light Waves ◽  
Hamilton Jacobi Equation ◽  
Common Action

Hamilton’s genius was to understand what were the true variables of mechanics (the “p − q,” conjugate coordinates, or canonical variables), and this led to Hamilton’s Mechanics which could obtain qualitative answers to a wider ranger of problems than Lagrangian Mechanics. It is explained how Hamilton’s canonical equations arise, why the Hamiltonian is the “central conception of all modern theory” (quote of Schrödinger’s), what the “p − q” variables are, and what phase space is. It is also explained how the famous conservation theorems arise (for energy, linear momentum, and angular momentum), and the connection with symmetry. The Hamilton-Jacobi Equation is derived using infinitesimal canonical transformations (ICTs), and predicts wavefronts of “common action” spreading out in (configuration) space. An analogy can be made with geometrical optics and Huygen’s Principle for the spreading out of light waves. It is shown how Hamilton’s Mechanics can lead into quantum mechanics.

scholarly journals Classification of extinction profiles for a one-dimensional diffusive Hamilton–Jacobi equation with critical absorption

2018 ◽  
Vol 148 (3) ◽  
pp. 559-574
Author(s):  
Razvan Gabriel Iagar ◽  
Philippe Laurençot
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Jacobi Equation ◽  
Threshold Value ◽  
Monotonicity Property ◽  
Fast Diffusion ◽  
Extinction Time ◽  
One Step ◽  
Hamilton Jacobi Equation

A classification of the behaviour of the solutions f(·, a) to the ordinary differential equation (|f′|p-2f′)′ + f - |f′|p-1 = 0 in (0,∞) with initial condition f(0, a) = a and f′(0, a) = 0 is provided, according to the value of the parameter a > 0 when the exponent p takes values in (1, 2). There is a threshold value a* that separates different behaviours of f(·, a): if a > a*, then f(·, a) vanishes at least once in (0,∞) and takes negative values, while f(·, a) is positive in (0,∞) and decays algebraically to zero as r→∞ if a ∊ (0, a*). At the threshold value, f(·, a*) is also positive in (0,∞) but decays exponentially fast to zero as r→∞. The proof of these results relies on a transformation to a first-order ordinary differential equation and a monotonicity property with respect to a > 0. This classification is one step in the description of the dynamics near the extinction time of a diffusive Hamilton–Jacobi equation with critical gradient absorption and fast diffusion.

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