3,311 research outputs found
Fractional Dirac Bracket and Quantization for Constrained Systems
So far, it is not well known how to deal with dissipative systems. There are
many paths of investigation in the literature and none of them present a
systematic and general procedure to tackle the problem. On the other hand, it
is well known that the fractional formalism is a powerful alternative when
treating dissipative problems. In this paper we propose a detailed way of
attacking the issue using fractional calculus to construct an extension of the
Dirac brackets in order to carry out the quantization of nonconservative
theories through the standard canonical way. We believe that using the extended
Dirac bracket definition it will be possible to analyze more deeply gauge
theories starting with second-class systems.Comment: Revtex 4.1. 9 pages, two-column. Final version to appear in Physical
Review
Between Laws and Models: Some Philosophical Morals of Lagrangian Mechanics
I extract some philosophical morals from some aspects of Lagrangian
mechanics. (A companion paper will present similar morals from Hamiltonian
mechanics and Hamilton-Jacobi theory.) One main moral concerns methodology:
Lagrangian mechanics provides a level of description of phenomena which has
been largely ignored by philosophers, since it falls between their accustomed
levels--``laws of nature'' and ``models''. Another main moral concerns
ontology: the ontology of Lagrangian mechanics is both more subtle and more
problematic than philosophers often realize.
The treatment of Lagrangian mechanics provides an introduction to the subject
for philosophers, and is technically elementary. In particular, it is confined
to systems with a finite number of degrees of freedom, and for the most part
eschews modern geometry. But it includes a presentation of Routhian reduction
and of Noether's ``first theorem''.Comment: 106 pages, no figure
Poisson integrators
An overview of Hamiltonian systems with noncanonical Poisson structures is
given. Examples of bi-Hamiltonian ode's, pde's and lattice equations are
presented. Numerical integrators using generating functions, Hamiltonian
splitting, symplectic Runge-Kutta methods are discussed for Lie-Poisson systems
and Hamiltonian systems with a general Poisson structure. Nambu-Poisson systems
and the discrete gradient methods are also presented.Comment: 30 page
Discrete Hamilton-Jacobi Theory
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.Comment: 26 pages, 2 figure
Characteristics, Bicharacteristics, and Geometric Singularities of Solutions of PDEs
Many physical systems are described by partial differential equations (PDEs).
Determinism then requires the Cauchy problem to be well-posed. Even when the
Cauchy problem is well-posed for generic Cauchy data, there may exist
characteristic Cauchy data. Characteristics of PDEs play an important role both
in Mathematics and in Physics. I will review the theory of characteristics and
bicharacteristics of PDEs, with a special emphasis on intrinsic aspects, i.e.,
those aspects which are invariant under general changes of coordinates. After a
basically analytic introduction, I will pass to a modern, geometric point of
view, presenting characteristics within the jet space approach to PDEs. In
particular, I will discuss the relationship between characteristics and
singularities of solutions and observe that: "wave-fronts are characteristic
surfaces and propagate along bicharacteristics". This remark may be understood
as a mathematical formulation of the wave/particle duality in optics and/or
quantum mechanics. The content of the paper reflects the three hour minicourse
that I gave at the XXII International Fall Workshop on Geometry and Physics,
September 2-5, 2013, Evora, Portugal.Comment: 26 pages, short elementary review submitted for publication on the
Proceedings of XXII IFWG
Discrete Hamiltonian Variational Integrators
We consider the continuous and discrete-time Hamilton's variational principle
on phase space, and characterize the exact discrete Hamiltonian which provides
an exact correspondence between discrete and continuous Hamiltonian mechanics.
The variational characterization of the exact discrete Hamiltonian naturally
leads to a class of generalized Galerkin Hamiltonian variational integrators,
which include the symplectic partitioned Runge-Kutta methods. We also
characterize the group invariance properties of discrete Hamiltonians which
lead to a discrete Noether's theorem.Comment: 23 page
Involutive constrained systems and Hamilton-Jacobi formalism
In this paper, we study singular systems with complete sets of involutive
constraints. The aim is to establish, within the Hamilton-Jacobi theory, the
relationship between the Frobenius' theorem, the infinitesimal canonical
transformations generated by constraints in involution with the Poisson
brackets, and the lagrangian point (gauge) transformations of physical systems
Parcel Eulerian-Lagrangian fluid dynamics for rotating geophysical flows
Parcel Eulerian-Lagrangian Hamiltonian formulations have recently been used in structure-preserving numerical schemes, asymptotic calculations and in alternative explanations of fluid parcel (in) stabilities. A parcel formulation describes the dynamics of one fluid parcel with a Lagrangian kinetic energy but an Eulerian potential evaluated at the parcel's position. In this paper, we derive the geometric link between the parcel Eulerian-Lagrangian formulation and well-known variational and Hamiltonian formulations for three models of ideal and geophysical fluid flow: generalized two-dimensional vorticity-stream function dynamics, the rotating two-dimensional shallow-water equations and the rotating three-dimensional compressible Euler equations
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