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The possibility of deforming the (associative or Lie) product to obtain alternative descriptions for a given classical or quantum system has been considered in many papers. Here we discuss the possibility of obtaining some novel alternative…
The conditions under which a given manifold $M$ may be given a tangent bundle or a cotangent bundle structure are analyzed. This is an important property arising in different contexts. For instance, in the study of integrability of a given…
The purpose of this paper is describe Lagrangian Mechanics for constrained systems on Lie algebroids, a natural framework which covers a wide range of situations (systems on Lie groups, quotients by the action of a Lie group, standard…
In describing a dynamical system, the greatest part of the work for a theoretician is to translate experimental data into differential equations. It is desirable for such differential equations to admit a Lagrangian and/or an Hamiltonian…
A path-integral quantization method is proposed for dynamical systems whose classical equations of motion do \textit{not} necessarily follow from the action principle. The key new notion behind this quantization scheme is the Lagrange…
This paper presents a geometric description on Lie algebroids of Lagrangian systems subject to nonholonomic constraints. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the…
The study of mechanical systems on Lie algebroids permits an understanding of the dynamics described by a Lagrangian or Hamiltonian function for a wide range of mechanical systems in a unified framework. Systems defined in tangent bundles,…
We study the reduction of non-autonomous regular Lagrangian systems by symmetries, which are generated by vector fields associated with connections in the configuration bundle of the system $Q\times\real\to\real$. These kind of symmetries…
We present a formulation of classical statistical mechanics based on a Lagrangian description on the tangent bundle. In this approach, a Wick rotation from real time to imaginary time is employed as a technical device that facilitates the…
A geometric approach to Sundman transformation defined by basic functions for systems of second-order differential equations is developed and the necessity of a change of the tangent structure by means of the function defining the Sundman…
This work introduces the concept of tangent space regularization for neural-network models of dynamical systems. The tangent space to the dynamics function of many physical systems of interest in control applications exhibits useful…
In fundamental theories that accounts for quantum gravitational effects, the spacetime causal structure is expected to be quantum uncertain. Previous studies of quantum causal structure focused on finite-dimensional systems. Here we present…
It is well-known that the conformal structure of a relativistic spacetime is of profound physical and conceptual interest. In this note, we consider the analogous structure for Newtonian theories. We show that the Newtonian Weyl tensor is…
We discuss examples of systems which can be quantized consistently, although they do not admit a Lagrangian description.
The thesis is devoted to the phase space representation of relativistic quantum mechanics. For a class of observables with matrix-valued Weyl symbols proportional to the identity matrix, the Weyl-Wigner-Moyal formalism is proposed. The…
In this work is discussed possibility and actuality of Lagrangian approach to quantum computations. Finite-dimensional Hilbert spaces used in this area provide some challenge for such consideration. The model discussed here can be…
Unitary representations of the Galilei group are studied in phase space, in order to describe classical and quantum systems. Conditions to write in general form the generator of time translation and Lagrangians in phase space are then…
A reductive structure is associated here with Lagrangian canonically defined conserved quantities on gauge-natural bundles. Parametrized transformations defined by the gauge-natural lift of infinitesimal principal automorphisms induce a…
The mathematical structure of the temporal gauge of QED is critically examined in both the alternative formulations characterized by either positivity or regularity of the Weyl algebra. The conflict between time translation invariance and…
The characterization of systems of differential equations admitting a superposition function allowing us to write the general solution in terms of any fundamental set of particular solutions is discussed. These systems are shown to be…