1,573 research outputs found
Geometrical dissipation for dynamical systems
On a Riemannian manifold we consider the functions
and construct the vector fields that conserve and
dissipate with a prescribed rate. We study the geometry of these vector
fields and prove that they are of gradient type on regular leaves corresponding
to . By using these constructions we show that the cubic Morrison
dissipation and the Landau-Lifschitz equation can be formulated in a unitary
form
Control of finite-dimensional quantum systems: Application to a spin-1/2 particle coupled with a finite quantum harmonic oscillator
In this paper, we consider the problem of the controllability of a finite-dimensional quantum system in both the Schrödinger and interaction pictures. Introducing a Quantum Transfer Graph, we elucidate the role of Lie algebra rank conditions and the complex nature of the control matrices. We analyze the example of a sequentially coupled N-level system: a spin-1/2 particle coupled to a finite quantum harmonic oscillator. This models an important physical paradigm of quantum computers - the trapped ion. We describe the control of the finite model obtained, under the right conditions, from the original infinite-dimensional system
Hamiltonization of Nonholonomic Systems and the Inverse Problem of the Calculus of Variations
We introduce a method which allows one to recover the equations of motion of
a class of nonholonomic systems by finding instead an unconstrained Hamiltonian
system on the full phase space, and to restrict the resulting canonical
equations to an appropriate submanifold of phase space. We focus first on the
Lagrangian picture of the method and deduce the corresponding Hamiltonian from
the Legendre transformation. We illustrate the method with several examples and
we discuss its relationship to the Pontryagin maximum principle.Comment: 23 pages, accepted for publication in Rep. Math. Phy
A Generalization of Chaplygin's Reducibility Theorem
In this paper we study Chaplygin's Reducibility Theorem and extend its
applicability to nonholonomic systems with symmetry described by the
Hamilton-Poincare-d'Alembert equations in arbitrary degrees of freedom. As
special cases we extract the extension of the Theorem to nonholonomic Chaplygin
systems with nonabelian symmetry groups as well as Euler-Poincare-Suslov
systems in arbitrary degrees of freedom. In the latter case, we also extend the
Hamiltonization Theorem to nonholonomic systems which do not possess an
invariant measure. Lastly, we extend previous work on conditionally variational
systems using the results above. We illustrate the results through various
examples of well-known nonholonomic systems.Comment: 27 pages, 3 figures, submitted to Reg. and Chaotic Dy
Moving constraints as stabilizing controls in classical mechanics
The paper analyzes a Lagrangian system which is controlled by directly
assigning some of the coordinates as functions of time, by means of
frictionless constraints. In a natural system of coordinates, the equations of
motions contain terms which are linear or quadratic w.r.t.time derivatives of
the control functions. After reviewing the basic equations, we explain the
significance of the quadratic terms, related to geodesics orthogonal to a given
foliation. We then study the problem of stabilization of the system to a given
point, by means of oscillating controls. This problem is first reduced to the
weak stability for a related convex-valued differential inclusion, then studied
by Lyapunov functions methods. In the last sections, we illustrate the results
by means of various mechanical examples.Comment: 52 pages, 4 figure
Discrete Variational Optimal Control
This paper develops numerical methods for optimal control of mechanical
systems in the Lagrangian setting. It extends the theory of discrete mechanics
to enable the solutions of optimal control problems through the discretization
of variational principles. The key point is to solve the optimal control
problem as a variational integrator of a specially constructed
higher-dimensional system. The developed framework applies to systems on
tangent bundles, Lie groups, underactuated and nonholonomic systems with
symmetries, and can approximate either smooth or discontinuous control inputs.
The resulting methods inherit the preservation properties of variational
integrators and result in numerically robust and easily implementable
algorithms. Several theoretical and a practical examples, e.g. the control of
an underwater vehicle, will illustrate the application of the proposed
approach.Comment: 30 pages, 6 figure
Equivalence of the Siegert-pseudostate and Lagrange-mesh R-matrix methods
Siegert pseudostates are purely outgoing states at some fixed point expanded
over a finite basis. With discretized variables, they provide an accurate
description of scattering in the s wave for short-range potentials with few
basis states. The R-matrix method combined with a Lagrange basis, i.e.
functions which vanish at all points of a mesh but one, leads to simple
mesh-like equations which also allow an accurate description of scattering.
These methods are shown to be exactly equivalent for any basis size, with or
without discretization. The comparison of their assumptions shows how to
accurately derive poles of the scattering matrix in the R-matrix formalism and
suggests how to extend the Siegert-pseudostate method to higher partial waves.
The different concepts are illustrated with the Bargmann potential and with the
centrifugal potential. A simplification of the R-matrix treatment can usefully
be extended to the Siegert-pseudostate method.Comment: 19 pages, 1 figur
Iso-spectral deformations of general matrix and their reductions on Lie algebras
We study an iso-spectral deformation of general matrix which is a natural
generalization of the Toda lattice equation. We prove the integrability of the
deformation, and give an explicit formula for the solution to the initial value
problem. The formula is obtained by generalizing the orthogonalization
procedure of Szeg\"{o}. Based on the root spaces for simple Lie algebras, we
consider several reductions of the hierarchy. These include not only the
integrable systems studied by Bogoyavlensky and Kostant, but also their
generalizations which were not known to be integrable before. The behaviors of
the solutions are also studied. Generically, there are two types of solutions,
having either sorting property or blowing up to infinity in finite time.Comment: 25 pages, AMSLaTe
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
- âŠ