Results 91 to 100 of about 336 (142)
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2013
In this chapter, we present the classical Hamilton-Jacobi theory. This theory has played an enormous role in the development of theoretical and mathematical physics. On the one hand, it builds a bridge between classical mechanics and other branches of physics, in particular, optics.
Gerd Rudolph, Matthias Schmidt
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In this chapter, we present the classical Hamilton-Jacobi theory. This theory has played an enormous role in the development of theoretical and mathematical physics. On the one hand, it builds a bridge between classical mechanics and other branches of physics, in particular, optics.
Gerd Rudolph, Matthias Schmidt
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2002
In the preceding chapter, we tried to perform a transformation to coordinate pairs (q i ,p i =β i ) for which the canonical momenta were constant. We now proceed one step further and look for a canonical transformation to coordinates P i =p i0 and Q i =q i0 which all are constant and are given by the initial conditions.
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In the preceding chapter, we tried to perform a transformation to coordinate pairs (q i ,p i =β i ) for which the canonical momenta were constant. We now proceed one step further and look for a canonical transformation to coordinates P i =p i0 and Q i =q i0 which all are constant and are given by the initial conditions.
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Generalized Hamilton–Jacobi theories
Journal of Mathematical Physics, 1978The generalized momenta of a dynamical system with n degrees of freedom may be given a Clebsch representation in terms of independent scalar functions and gradients thereof. The canonical equations imply certain differential relations which must be satisfied by these scalars.
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2017
Besides the Newtonian, Lagrangian, and Hamiltonian formulations of classical mechanics, there is yet a fourth approach, known as Hamilton–Jacobi theory, which is part of Hamiltonian mechanics. This approach is the subject of the present chapter. In Hamilton–Jacobi theory, the central equation capturing the dynamics of the mechanical system is the ...
Vicente Cortés, Alexander S. Haupt
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Besides the Newtonian, Lagrangian, and Hamiltonian formulations of classical mechanics, there is yet a fourth approach, known as Hamilton–Jacobi theory, which is part of Hamiltonian mechanics. This approach is the subject of the present chapter. In Hamilton–Jacobi theory, the central equation capturing the dynamics of the mechanical system is the ...
Vicente Cortés, Alexander S. Haupt
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2016
The considerations of the last section of the last chapter concerning canonical transformations reveal such a manifold of transformation possibilities that it should in fact be possible to construct by it efficient general methods for the solution of mechanical problems We therefore investigate now which way a Hamilton function H has to be transformed ...
Anatoli Babin, Alexander Figotin
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The considerations of the last section of the last chapter concerning canonical transformations reveal such a manifold of transformation possibilities that it should in fact be possible to construct by it efficient general methods for the solution of mechanical problems We therefore investigate now which way a Hamilton function H has to be transformed ...
Anatoli Babin, Alexander Figotin
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1992
Abstract Eqn (4.1.1) is the Hamilton-Jacobi equation (more accurately, it is the time-independent form of the Hamilton-Jacobi equation). Although it was originally introduced as a device for obtaining analytic solutions to mechanical problems, it is, in fact, of very little use in that context since the only generally applicable method ...
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Abstract Eqn (4.1.1) is the Hamilton-Jacobi equation (more accurately, it is the time-independent form of the Hamilton-Jacobi equation). Although it was originally introduced as a device for obtaining analytic solutions to mechanical problems, it is, in fact, of very little use in that context since the only generally applicable method ...
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2014
Dieses Kapitel stellt eine enge Beziehung her zwischen den gewohnlichen Differentialgleichungen von Hamilton und einer partiellen Differentialgleichung von Hamilton und Jacobi. Diese spielte, neben den Anwendungen in der Mechanik, beispielsweise auch eine wichtige Rolle in Schrodingers Suche nach seiner Wellengleichung.
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Dieses Kapitel stellt eine enge Beziehung her zwischen den gewohnlichen Differentialgleichungen von Hamilton und einer partiellen Differentialgleichung von Hamilton und Jacobi. Diese spielte, neben den Anwendungen in der Mechanik, beispielsweise auch eine wichtige Rolle in Schrodingers Suche nach seiner Wellengleichung.
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1996
Ich komme nun schlieslich zu einer Form, in welcher das System der dynamischen Differentialgleichungen dargestellt werden kann, die das ganze Problem auf ein anderes Gebiet versetzt. Ich schicke hier folgende allgemeine Grundlagen voraus. Die Aufgaben der Variationsrechnung haben zu einer Betrachtung des Grosten und Kleinsten gefuhrt, wofur verlangt ...
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Ich komme nun schlieslich zu einer Form, in welcher das System der dynamischen Differentialgleichungen dargestellt werden kann, die das ganze Problem auf ein anderes Gebiet versetzt. Ich schicke hier folgende allgemeine Grundlagen voraus. Die Aufgaben der Variationsrechnung haben zu einer Betrachtung des Grosten und Kleinsten gefuhrt, wofur verlangt ...
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2005
Abstract The Hamilton-Jacobi theory is the apotheosis of Lagrangian and Hamiltonian mechanics: action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria. These action functions are the solutions of a nonlinear, first-order partial differential equation, called the Hamilton-Jacobi equation.
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Abstract The Hamilton-Jacobi theory is the apotheosis of Lagrangian and Hamiltonian mechanics: action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria. These action functions are the solutions of a nonlinear, first-order partial differential equation, called the Hamilton-Jacobi equation.
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2012
We have repeatedly emphasized that determining the explicit solutions of Hamiltonian equations is a task beyond the capabilities of mathematics. This situation justifies all attempts to obtain information about these solutions. For instance, a knowledge of the first integrals allows us to localize the dynamical trajectories in the phase space M 2n ...
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We have repeatedly emphasized that determining the explicit solutions of Hamiltonian equations is a task beyond the capabilities of mathematics. This situation justifies all attempts to obtain information about these solutions. For instance, a knowledge of the first integrals allows us to localize the dynamical trajectories in the phase space M 2n ...
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