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Moving Picture and Hamilton-Jacobi Theory in Quantum Mechanics  [PDF]
Akihiro Ogura,Motoo Sekiguchi
Physics , 2003,
Abstract: We propose a new picture, which we call the {\it moving picture}, in quantum mechanics. The Schr\"{o}dinger equation in this picture is derived and its solution is examined. We also investigate the close relationship between the moving picture and the Hamilton-Jacobi theory in classical mechanics.
Discrete Hamilton-Jacobi Theory  [PDF]
Tomoki Ohsawa,Anthony M. Bloch,Melvin Leok
Mathematics , 2009, DOI: 10.1137/090776822
Abstract: We develop a discrete analogue of Hamilton-Jacobi theory in the framework of discrete Hamiltonian mechanics. The resulting discrete Hamilton-Jacobi equation is discrete only in time. We describe a discrete analogue of Jacobi's solution and also prove a discrete version of the geometric Hamilton-Jacobi theorem. The theory applied to discrete linear Hamiltonian systems yields the discrete Riccati equation as a special case of the discrete Hamilton-Jacobi equation. We also apply the theory to discrete optimal control problems, and recover some well-known results, such as the Bellman equation (discrete-time HJB equation) of dynamic programming and its relation to the costate variable in the Pontryagin maximum principle. This relationship between the discrete Hamilton-Jacobi equation and Bellman equation is exploited to derive a generalized form of the Bellman equation that has controls at internal stages.
A Universal Hamilton-Jacobi Theory  [PDF]
Manuel de León,David Martín de Diego,Miguel Vaquero
Mathematics , 2012,
Abstract: In this paper we develop a Hamilton-Jacobi theory in the setting of almost Poisson manifolds. The theory extends the classical Hamilton-Jacobi theory and can be also applied to very general situations including nonholonomic mechanical systems and time dependent systems with external forces.
Hamilton-Jacobi theory in Cauchy data space  [PDF]
Cédirc M. Campos,Manuel de León,David Martín de Diego,Miguel Vaquero
Physics , 2014,
Abstract: Recently, M. de Le\'on el al. ([9]) have developed a geometric Hamilton-Jacobi theory for Classical Field Theories in the setting of multisymplectic geometry. Our purpose in the current paper is to establish the corresponding Hamilton-Jacobi theory in the Cauchy data space, and relate both approaches.
Quantum Hamilton-Jacobi Theory  [PDF]
Marco Roncadelli,L. S. Schulman
Physics , 2007, DOI: 10.1103/PhysRevLett.99.170406
Abstract: Quantum canonical transformations have attracted interest since the beginning of quantum theory. Based on their classical analogues, one would expect them to provide a powerful quantum tool. However, the difficulty of solving a nonlinear operator partial differential equation such as the quantum Hamilton-Jacobi equation (QHJE) has hindered progress along this otherwise promising avenue. We overcome this difficulty. We show that solutions to the QHJE can be constructed by a simple prescription starting from the propagator of the associated Schroedinger equation. Our result opens the possibility of practical use of quantum Hamilton-Jacobi theory. As an application we develop a surprising relation between operator ordering and the density of paths around a semiclassical trajectory.
Geometric Hamilton-Jacobi Field Theory  [PDF]
Luca Vitagliano
Mathematics , 2011, DOI: 10.1142/S0219887812600080
Abstract: I briefly review my proposal about how to extend the geometric Hamilton-Jacobi theory to higher derivative field theories on fiber bundles.
On the Hamilton-Jacobi Theory for Singular Lagrangian Systems  [PDF]
Manuel de León,Juan Carlos Marrero,David Martín de Diego,Miguel Vaquero
Physics , 2012, DOI: 10.1063/1.4796088
Abstract: We develop a Hamilton-Jacobi theory for singular lagrangian systems using the Gotay-Nester-Hinds constraint algorithm. The procedure works even if the system has secondary constraints.
Hamilton-Jacobi theory, Symmetries and Coisotropic Reduction  [PDF]
Manuel de León,David Martín de Diego,Miguel Vaquero
Mathematics , 2015,
Abstract: Reduction theory has played a major role in the study of Hamiltonian systems. On the other hand, the Hamilton-Jacobi theory is one of the main tools to integrate the dynamics of certain Hamiltonian problems and a topic of research on its own. Moreover, the construction of several symplectic integrators rely on approximations of a complete solution of the Hamilton-Jacobi equation. The natural question that we address in this paper is how these two topics (reduction and Hamilton-Jacobi theory) fit together. We obtain a reduction and reconstruction procedure for the Hamilton-Jacobi equation with symmetries, even in a generalized sense to be clarified below. Several applications and relations to other reductions of the Hamilton-Jacobi theory are shown in the last section of the paper. It is remarkable that as a by-product we obtain a generalization of the Ge-Marsden reduction procedure. Quite surprinsingly, the classical ansatzs available in the literature to solve the Hamilton-Jacobi equation are also particular instances of our framework.
Hamilton-Jacobi theory and the evolution operator  [PDF]
J. F. Carinena,X. Gracia,E. Martinez,G. Marmo,M. C. Munoz-Lecanda,N. Roman-Roy
Physics , 2009,
Abstract: We present a new setting of the geometric Hamilton-Jacobi theory by using the so-called time-evolution operator K. This new approach unifies both the Lagrangian and the Hamiltonian formulation of the problem developed in a previous paper [7], and can be applied to the case of singular Lagrangian dynamical systems.
Incompleteness of the Hamilton-Jacobi theory  [PDF]
Nivaldo A. Lemos
Physics , 2013, DOI: 10.1119/1.4876355
Abstract: The problem of the motion of a charged particle in an electric dipole field is used to illustrate that the Hamilton-Jacobi method does not necessarily give all solutions to the equations of motion of a mechanical system. The mathematical reason for this phenomenon is discussed. In the particular case under consideration, it is shown how to circumvent the difficulty and find the missing solutions by means of a very special limiting procedure.
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