Hamiltonian Formulation for Continuous Systems with Second-Order Derivatives: A Study of Podolsky Generalized Electrodynamics (original) (raw)

The Hamiltonian Formulation of Higher Order Dynamical Systems

1994

Using Dirac's approach to constrained dynamics, the Hamiltonian formulation of regular higher order Lagrangians is developed. The conventional description of such systems due to Ostrogradsky is recovered. However, unlike the latter, the present analysis yields in a transparent manner the local structure of the associated phase space and its local sympletic geometry, and is of direct application to {\em constrained\/}

A reformulation of mechanics and electrodynamics

Heliyon, 2017

Classical mechanics, as commonly taught in engineering and science, are confined to the conventional Newtonian theory. But classical mechanics has not really changed in substance since Newton formulation, describing simultaneous rotation and translation of objects with somewhat complicate drawbacks, risking interpretation of forces in non-inertial frames. In this work we introduce a new variational principle for out-of-equilibrium, rotating systems, obtaining a set of two first order differential equations that introduces a thermodynamic-mechanistic time into Newton's dynamical equation, and revealing the same formal symplectic structure shared by classical mechanics, fluid mechanics and thermodynamics. The results is a more consistent formulation of dynamics and electrodynamics, explaining natural phenomena as the outcome from a balance between energy and entropy, embedding translational with rotational motion into a single equation, showing centrifugal and Coriolis force as derivatives from the transport of angular momentum, and offering a natural method to handle variational problems, as shown with the brachistochrone problem. In consequence, a new force term appears, the topological torsion current, important for spacecraft dynamics. We describe a set of solved problems showing the potential of a competing technique, with significant interest to electrodynamics as well. We expect this new approach to have impact in a large class of scientific and technological problems.

Dirac and Lagrange Algebraic Constraints in Nonlinear Port-Hamiltonian Systems

Vietnam Journal of Mathematics

After recalling the definitions of standard port-Hamiltonian systems and their algebraic constraints, called here Dirac algebraic constraints, an extended class of port-Hamiltonian systems is introduced. This is based on replacing the Hamiltonian function by a general Lagrangian submanifold of the cotangent bundle of the state space manifold, motivated by developments in (Barbero-Linan et al., J. Geom. Mech. 11, 487-510, 2019) and extending the linear theory as developed in (van der Schaft and Maschke, Syst. Control Lett. 121, 31-37, 2018) and (Beattie et al., Math. Control Signals Syst. 30, 17, 2018). The resulting new type of algebraic constraints equations are called Lagrange algebraic constraints. It is shown how Dirac algebraic constraints can be converted into Lagrange algebraic constraints by the introduction of extra state variables, and, conversely, how Lagrange algebraic constraints can be converted into Dirac algebraic constraints by the use of Morse families.

On Hamiltonian Formulations of the Schrödinger System

Annals of Physics, 2002

We review and compare different variational formulations for the Schrödinger field. Some of them rely on the addition of a conveniently chosen total time derivative to the hermitic Lagrangian. Alternatively, the Dirac-Bergmann algorithm yields the Schrödinger equation first as a consistency condition in the full phase space, second as canonical equation in the reduced phase space. The two methods lead to the same (reduced) Hamiltonian. As a third possibility, the Faddeev-Jackiw method is shown to be a shortcut of the Dirac method. By implementing the quantization scheme for systems with second class constraints, inconsistencies of previous treatments are eliminated.

Fractional Quantization of Podolsky Electrodynamics Using Fractional Hamilton-Jacobi Formulation

NSp, 2023

For fractional derivative order constrained systems, the Hamilton-Jacobi formulation in terms Riemann-Liouville fractional derivative was developed. The equations of motion are written as total differential fractional equations fractional in many variables using this formalism. We use the Hamilton-Jacobi formulation in terms of Riemann-Liouville fractional derivative to study Podolsky electrodynamics, comparing our results to those obtained using the Euler-Lagrange Riemann-Liouville fractional derivative method. A fractional difference will be presented as a minor adjustment to a Hamilton-Jacobi derivation formula that is more compatible with the traditional similarity. After generalizing Podolsky electrodynamics for constrained systems with fractional second-order Lagrangians, a new formulation is used to help the reader understand the conclusions.

On Hamiltonian formulation of non-conservative systems

Turk. J. Phys, 2004

Fractional derivatives are used to construct the Lagrangian and the Hamiltonian formulation for nonconservative systems. To clarify the theory of Riewe two interesting examples are given. The potentials are obtained using the Laplace transform operator for fractional derivatives and the Lagrangian and Hamiltonian formulations are constructed for the two systems. Besides, it is shown that the Hamilton equations of motion are in agreement with the Euler-Lagrange equations for these systems.

A matrix electrodynamics as an analogue of the Heisenberg's mechanics

2008 8th International Symposium on Antennas, Propagation and EM Theory, 2008

A matrix approach to solving the electrodynamic problems is suggested. The specificity of one is treatment of an electrodynamic system (ES) as an oscillating system with a finite number of the degrees of freedom. The ES is considered as a set of spatially localized so-called partial oscillators (oscillets). Matrices of unit mutual pseudoenergies and unit mutual energies of the oscillators are evaluated. The eigenfrequencies and the eigenfunctions of the ES can be calculated basing on the lumped elements oscillating system matrix theory. A matrix second-order ordinary differential equation is solved for excited potentials of the ES instead of the D'Alembert equation. The main advantage of the matrix electrodynamics is substitution of the solving the partial derivative differential equations by the less computationally intensive linear algebra problems and the ordinary differential equation integration.

The Lagrangian and Hamiltonian formalisms for the classical relativistic electrodynamics models revisited

The work is devoted to studying some new classical electrodynamics models of interacting charged point particles and the aspects of the quantization via the Dirac procedure related to them. Based on the vacuum field theory no-geometry approach developed in [6,7,9], the Lagrangian and Hamiltonian reformulations of some alternative classical electrodynamics models are devised. The Dirac-type quantization procedure for the considered alternative electrodynamics models, based on the obtained canonical Hamiltonian formulations, is developed. Comment: 9 pages

Higher order complex Lagrangian and Hamiltonian mechanics systems

Physics Letters A, 2006

In this Letter, it was present higher order vertical and complete lifts of Euler-Lagrange and Hamiltonian equations introduced on Kählerian manifold to its extensions. Finally, it was discussed differential geometric and physical results for the higher order Lagrangian and Hamiltonian mechanical systems obtained on the extended Kählerian manifolds.

Application of Generalized Hamiltonian Dynamics to Modified Coulomb Potential

Aps Texas Sections Fall Meeting Abstracts, 2008

Dirac's Generalized Hamiltonian Dynamics (GHD), a purely classical formalism, is applied to spinless particles under the influence of a binomial potential. The integrals of the motion for this potential are chosen as the constraints of GHD and Fradkin's unit Runge vector is used in place of the Laplace-Runge-Lenz vector. A functional form of the unit Runge vector is derived for the binomial potential. It is shown-in accordance with Oks and Uzer (2002)that there occurs a new kind of time dilation leading to classical stable, nonradiating states. The energy of these classical stable states agrees exactly with the corresponding quantal results for the ground state and for all states of odd values of the radial and angular harmonic numbers. The primary application of the obtained results is to pionic (and kaonic) atoms. Other applications include nanoplasmas and the precession of planetary orbits.