Some applications of quasi-velocities in optimal control

In this paper we study optimal control problems for nonholonomic systems defined on Lie algebroids by using quasi-velocities. We consider both kinematic, i.e. systems whose cost functional depends only on position and velocities, and dynamic optimal control problems, i.e. systems whose cost functional depends also on accelerations. The formulation of the problem directly at the level of Lie algebroids turns out to be the correct framework to explain in detail similar results appeared recently (Maruskin and Bloch, 2007). We also provide several examples to illustrate our construction.Comment: Revtex 4.1, 20 pages. To appear in Int. J. Geom. Meth. Modern Physic

Time Evolution In Macroscopic Systems. II: The Entropy

The concept of entropy in nonequilibrium macroscopic systems is investigated in the light of an extended equation of motion for the density matrix obtained in a previous study. It is found that a time-dependent information entropy can be defined unambiguously, but it is the time derivative or entropy production that governs ongoing processes in these systems. The differences in physical interpretation and thermodynamic role of entropy in equilibrium and nonequilibrium systems is emphasized and the observable aspects of entropy production are noted. A basis for nonequilibrium thermodynamics is also outlinedComment: 28 page

Statistical thermodynamics for a non-commutative special relativity: Emergence of a generalized quantum dynamics

There ought to exist a description of quantum field theory which does not depend on an external classical time. To achieve this goal, in a recent paper we have proposed a non-commutative special relativity in which space-time and matter degrees of freedom are treated as classical matrices with arbitrary commutation relations, and a space-time line element is defined using a trace. In the present paper, following the theory of Trace Dynamics, we construct a statistical thermodynamics for the non-commutative special relativity, and show that one arrives at a generalized quantum dynamics in which space and time are non-classical and have an operator status. In a future work, we will show how standard quantum theory on a classical space-time background is recovered from here.Comment: 21 pages. arXiv admin note: text overlap with arXiv:1106.091

Schwinger's Principle and Gauge Fixing in the Free Electromagnetic Field

A manifestly covariant treatment of the free quantum eletromagnetic field, in a linear covariant gauge, is implemented employing the Schwinger's Variational Principle and the B-field formalism. It is also discussed the abelian Proca's model as an example of a system without constraints.Comment: 8 pages. Format PTPtex. No figur

Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte

Dissipative Time Evolution of Observables in Non-equilibrium Statistical Quantum Systems

We discuss differential-- versus integral--equation based methods describing out--of thermal equilibrium systems and emphasize the importance of a well defined reduction to statistical observables. Applying the projection operator approach, we investigate on the time evolution of expectation values of linear and quadratic polynomials in position and momentum for a statistical anharmonic oscillator with quartic potential. Based on the exact integro-differential equations of motion, we study the first and naive second order approximation which breaks down at secular time-scales. A method is proposed to improve the expansion by a non--perturbative resummation of all quadratic operator correlators consistent with energy conservation for all times. Motion cannot be described by an effective Hamiltonian local in time reflecting non-unitarity of the dissipative entropy generating evolution. We numerically integrate the consistently improved equations of motion for large times. We relate entropy to the uncertainty product, both being expressible in terms of the observables under consideration.Comment: 20 pages, 6 Figure

Quantum Geometrodynamics I: Quantum-Driven Many-Fingered Time

The classical theory of gravity predicts its own demise -- singularities. We therefore attempt to quantize gravitation, and present here a new approach to the quantization of gravity wherein the concept of time is derived by imposing the constraints as expectation-value equations over the true dynamical degrees of freedom of the gravitational field -- a representation of the underlying anisotropy of space. This self-consistent approach leads to qualitatively different predictions than the Dirac and the ADM quantizations, and in addition, our theory avoids the interpretational conundrums associated with the problem of time in quantum gravity. We briefly describe the structure of our functional equations, and apply our quantization technique to two examples so as to illustrate the basic ideas of our approach.Comment: 11, (No Figures), (Typeset using RevTeX