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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.
The stochastic Hamilton-Jacobi equation  [PDF]
Joan-Andreu Lázaro-Camí,Juan-Pablo Ortega
Mathematics , 2008,
Abstract: We extend some aspects of the Hamilton-Jacobi theory to the category of stochastic Hamiltonian dynamical systems. More specifically, we show that the stochastic action satisfies the Hamilton-Jacobi equation when, as in the classical situation, it is written as a function of the configuration space using a regular Lagrangian submanifold. Additionally, we will use a variation of the Hamilton-Jacobi equation to characterize the generating functions of one-parameter groups of symplectomorphisms that allow to rewrite a given stochastic Hamiltonian system in a form whose solutions are very easy to find; this result recovers in the stochastic context the classical solution method by reduction to the equilibrium of a Hamiltonian system.
Quantum Hamilton-Jacobi equation  [PDF]
Vipul Periwal
Physics , 1997, DOI: 10.1103/PhysRevLett.80.4366
Abstract: The nontrivial transformation of the phase space path integral measure under certain discretized analogues of canonical transformations is computed. This Jacobian is used to derive a quantum analogue of the Hamilton-Jacobi equation for the generating function of a canonical transformation that maps any quantum system to a system with a vanishing Hamiltonian. A formal perturbative solution of the quantum Hamilton-Jacobi equation is given.
Hamilton-Jacobi Theory and Moving Frames  [cached]
Joshua D. MacArthur,Raymond G. McLenaghan,Roman G. Smirnov
Symmetry, Integrability and Geometry : Methods and Applications , 2007,
Abstract: The interplay between the Hamilton-Jacobi theory of orthogonal separation of variables and the theory of group actions is investigated based on concrete examples.
Hamilton-Jacobi Theory and Moving Frames  [PDF]
Joshua D. MacArthur,Raymond G. McLenaghan,Roman G. Smirnov
Physics , 2007, DOI: 10.3842/SIGMA.2007.070
Abstract: The interplay between the Hamilton-Jacobi theory of orthogonal separation of variables and the theory of group actions is investigated based on concrete examples.
The Teleparallel Lagrangian and Hamilton-Jacobi Formalism  [PDF]
B. M. Pimentel,P. J. Pompeia,J. F. da Rocha-Neto,R. G. Teixeira
Physics , 2003, DOI: 10.1023/A:1022951321978
Abstract: We analyze the Teleparallel Equivalent of General Relativity (TEGR) from the point of view of Hamilton-Jacobi approach for singular systems
The Hamilton-Jacobi equation on Lie affgebroids  [PDF]
Juan Carlos Marrero,Diana Sosa
Mathematics , 2005,
Abstract: The Hamilton-Jacobi equation for a Hamiltonian section on a Lie affgebroid is introduced and some examples are discussed.
The Hamilton-Jacobi Approach to Teleparallelism  [PDF]
B. M. Pimentel,P. J. Pompeia,J. F. da Rocha-Neto
Physics , 2004, DOI: 10.1393/ncb/i2005-10085-4
Abstract: We intend to analyse the constraint structure of Teleparallelism employing the Hamilton-Jacobi formalism for singular systems. This study is conducted without using an ADM 3+1 decomposition and without fixing time gauge condition. It can be verified that the field equations constitute an integrable system.
Generalized Hamilton-Jacobi equations for nonholonomic dynamics  [PDF]
Michele Pavon
Physics , 2004, DOI: 10.1063/1.1858441
Abstract: Employing a suitable nonlinear Lagrange functional, we derive generalized Hamilton-Jacobi equations for dynamical systems subject to linear velocity constraints. As long as a solution of the generalized Hamilton-Jacobi equation exists, the action is actually minimized (not just extremized).
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.
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