36 research outputs found
The nonlinear superposition principle and the Wei-Norman method
Group theoretical methods are used to study some properties of the Riccati
equation, which is the only differential equation admitting a nonlinear
superposition principle. The Wei-Norman method is applied to obtain the
associated differential equation in the group . The superposition
principle for first order differential equation systems and Lie-Scheffers
theorem are also analysed from this group theoretical perspective. Finally, the
theory is applied in the solution of second order differential equations like
time-independent Schroedinger equatio
Contractions: Nijenhuis and Saletan tensors for general algebraic structures
Generalizations in many directions of the contraction procedure for Lie
algebras introduced by E.J.Saletan are proposed. Products of arbitrary nature,
not necessarily Lie brackets, are considered on sections of finite-dimensional
vector bundles. Saletan contractions of such infinite-dimensional algebras are
obtained via a generalization of the Nijenhuis tensor approach. In particular,
this procedure is applied to Lie algebras, Lie algebroids, and Poisson
structures. There are also results on contractions of n-ary products and
coproducts.Comment: 25 pages, LateX, corrected typo
Introduction to Quantum Mechanics and the Quantum-Classical transition
In this paper we present a survey of the use of differential geometric
formalisms to describe Quantum Mechanics. We analyze Schroedinger and
Heisenberg frameworks from this perspective and discuss how the momentum map
associated to the action of the unitary group on the Hilbert space allows to
relate both approaches. We also study Weyl-Wigner approach to Quantum Mechanics
and discuss the implications of bi-Hamiltonian structures at the quantum level.Comment: Survey paper based on the lectures delivered at the XV International
Workshop on Geometry and Physics Puerto de la Cruz, Tenerife, Canary Islands,
Spain September 11-16, 2006. To appear in Publ. de la RSM
The transfer matrix: a geometrical perspective
We present a comprehensive and self-contained discussion of the use of the
transfer matrix to study propagation in one-dimensional lossless systems,
including a variety of examples, such as superlattices, photonic crystals, and
optical resonators. In all these cases, the transfer matrix has the same
algebraic properties as the Lorentz group in a (2+1)-dimensional spacetime, as
well as the group of unimodular real matrices underlying the structure of the
abcd law, which explains many subtle details. We elaborate on the geometrical
interpretation of the transfer-matrix action as a mapping on the unit disk and
apply a simple trace criterion to classify the systems into three types with
very different geometrical and physical properties. This approach is applied to
some practical examples and, in particular, an alternative framework to deal
with periodic (and quasiperiodic) systems is proposed.Comment: 50 pages, 24 figure
Monopole-based quantization: a programme
We describe a programme to quantize a particle in the field of a (three
dimensional) magnetic monopole using a Weyl system. We propose using the
mapping of position and momenta as operators on a quaternionic Hilbert module
following the work of Emch and Jadczyk.Comment: Contribution to the volume: Mathematical Physics and Field Theory,
Julio Abad, In Memoriam}, M. Asorey, J.V. Garcia Esteve, M.F. Ranada and J.
Sesma Editors, Prensas Universitaria de Zaragoza, (2009
Geometric Hamilton-Jacobi Theory
The Hamilton-Jacobi problem is revisited bearing in mind the consequences
arising from a possible bi-Hamiltonian structure. The problem is formulated on
the tangent bundle for Lagrangian systems in order to avoid the bias of the
existence of a natural symplectic structure on the cotangent bundle. First it
is developed for systems described by regular Lagrangians and then extended to
systems described by singular Lagrangians with no secondary constraints. We
also consider the example of the free relativistic particle, the rigid body and
the electron-monopole system.Comment: 40 page
Classical and Quantum Systems: Alternative Hamiltonian Descriptions
In complete analogy with the classical situation (which is briefly reviewed)
it is possible to define bi-Hamiltonian descriptions for Quantum systems. We
also analyze compatible Hermitian structures in full analogy with compatible
Poisson structures.Comment: To appear on Theor. Math. Phy
Optimal path planning for nonholonomic robotics systems via parametric optimisation
Abstract. Motivated by the path planning problem for robotic systems this paper considers nonholonomic path planning on the Euclidean group of motions SE(n) which describes a rigid bodies path in n-dimensional Euclidean space. The problem is formulated as a constrained optimal kinematic control problem where the cost function to be minimised is a quadratic function of translational and angular velocity inputs. An application of the Maximum Principle of optimal control leads to a set of Hamiltonian vector field that define the necessary conditions for optimality and consequently the optimal velocity history of the trajectory. It is illustrated that the systems are always integrable when n = 2 and in some cases when n = 3. However, if they are not integrable in the most general form of the cost function they can be rendered integrable by considering special cases. This implies that it is possible to reduce the kinematic system to a class of curves defined analytically. If the optimal motions can be expressed analytically in closed form then the path planning problem is reduced to one of parameter optimisation where the parameters are optimised to match prescribed boundary conditions.This reduction procedure is illustrated for a simple wheeled robot with a sliding constraint and a conventional slender underwater vehicle whose velocity in the lateral directions are constrained due to viscous damping