Lie Group Framework Research Articles

Gradient flows and double bracket equations

A unified extension of the gradient flows and the double bracket equations of Chu–Driessel and Brockett is obtained in the frame work of reductive Lie groups. We examine the gradient flows on the orbit in the Cartan subspace of a reductive Lie algebra, under the adjoint action. The results of Chu–Driessel and Brockett are corresponding to the reductive groups GL(n, R) and O( p, q).

A differential geometric viewpoint on local identifiability and identification part I: theory

The questions of local identifiability and identification of nonlinear systems are treated from a differential geometric point of view. It is shown how identifiability can be interpreted in the framework of Lie groups, which provides convenient tools for a unifying concept of identifiability. Three different notions of identifiability are interpreted in this framework. Furthermore parameter identification itself is also treated. It is shown how a parameter estimator can be formulated in a coordinate-free setting. In this setting it can be clearly determined to what extent the underlying geometric structure and the coordinate-specific parameterization of the system influence the properties of the estimator.

Euclidean metrics for motion generation on SE(3)

Previous approaches to trajectory generation for rigid bodies have been either based on the so-called invariant screw motions or on ad hoc decompositions into rotations and translations. This paper formulates the trajectory generation problem in the framework of Lie groups and Riemannian geometry. The goal is to determine optimal curves joining given points with appropriate boundary conditions on the Euclidean group. Since this results in a two-point boundary value problem that has to be solved iteratively, a computationally efficient, analytical method that generates near-optimal trajectories is derived. The method consists of two steps. The first step involves generating the optimal trajectory in an ambient space, while the second step is used to project this trajectory onto the Euclidean group. The paper describes the method, its applications and its performance in terms of optimality and efficiency.

Vers une neurogéométrie. Fibrations corticales, structures de contact et contours subjectifs modaux

This work presents some variational models for the cortical algorithms processing Kanizsa modal subjective contours . These models are based on the geometric concepts of fibration and contact structure. The retinoptic structure of the orientation hypercolumns in the visual area V1 is a functionnal architecture which can be mathematically idealized by the fibration having the retinian plane M as base and the projective line P1 as fiber F. The total space E of Pi p is isomorphic to the direct product M x F. The cortico-cortical horizontal connections implement what is called the local triviality of this fibration, and also a Cartan connection defining a parallel transport between neighboring fibers. Then the paper focuses on the geometrical interpretation of the results of Field, Hayes and Hess concerning the association field. It shows that the latter implements what is called the contact structure of the fibration. The association field expresses an integrability condition for the skew curves in E : they have to be a lifting of their projection on the retinian plane M. This model of fibration endowed with a contact structure is then applied to the modal subjective contours and provides a variant of the elastica model developped by B.K.P. Horn and D. Mumford. The key idea is that the lifting of subjective contours satisfy a "geodesic" condition in the cortical fibration E : they have to be of minimal lenght (for an appropriate metrics) among the class of curves satisfying the integrability condition. These "geodesic" models are then reformulated, according to R. Bryant and P. Griffiths, in the more fondamental geometric framework of Lie groups and Cartan's "repère mobile" (Vielbein). Finally, some experimental possibilities are suggested.

An analytic approach to stochastic calculus

An Itô type stochastic differential calculus is constructed in an analytic way, without use of the notion of filtration, for processes that satisfy certain homogeneity and smoothness conditions. This calculus is developed in the framework of Lie groups.

Trasformate di Radon e operatori di convoluzione su gruppi e algebre di Lie

The following is an expository paper concerning the relations between Radon transforms and convolution operators associated to singular measures. A quick review of the classical theorems is presented and a recent result of F. Ricci and the author in the framework of compact Lie groups and Lie algebras is outlined.

Hamiltonian dynamics of a rigid body in a central gravitational field

This paper concerns the dynamics of a rigid body of finite extent moving under the influence of a central gravitational field. A principal motivation behind this paper is to reveal the hamiltonian structure of the n-body problem for masses of finite extent and to understand the approximation inherent to modeling the system as the motion of point masses. To this end, explicit account is taken of effects arising because of the finite extent of the moving body. In the spirit of Arnold and Smale, exact models of spin-orbit coupling are formulated, with particular attention given to the underlying Lie group framework. Hamiltonian structures associated with such models are carefully constructed and shown to benon-canonical. Special motions, namely relative equilibria, are investigated in detail and the notion of anon-great circle relative equilibrium is introduced. Non-great circle motions cannot arise in the point mass model. In our analysis, a variational characterization of relative equilibria is found to be very useful.

An illustration of the Lie group framework for soliton equations: Generalizations of the Lund–Regge model

The Lie group framework for soliton equations is illustrated. It is shown that the original Lund–Regge model is one of an infinite family of similar relativistically invariant models that possess associated eigenvalue problems and isospectral flows. The models are explicitly found and their associated structures displayed. The group theoretic significance of the soliton equations and associated structures are given in accordance with the general theory.

A Lie group framework for soliton equations. I. Path independent case

A general Lie group theoretic framework for the study of a class of nonlinear partial differential equations is presented. In two space–time dimensions this class includes soliton equations. The approach is applicable in N⩽2 space–time dimensions. Eigenvalue problems and isospectral flows associated with equations have a natural group theoretic interpretation in this framework. A sequence of nonlocal exact 1-,2-,...,(N−1) -forms are derived in N-dimensional space–time.

Symplectic symmetry of hadrons

A model is proposed forSU3 symmetry in which the knownSU3 multiplets of hadrons (strongly interacting particles) are considered as bound states of basic particles t of spin 1/2, baryon number 1 and electric charge 0, ±1, and of their antiparticles\(\bar t\). These basic particles formSU3 triplets and are regarded to be heavy compared to the proton mass. The known meson octets and singlet are supposed to result from binding of a t and a\(\bar t\), giving in terms ofSU3 symmetry the structure\(\bar 3\)×3=1+8. The baryon octet, decuplet (and possibly singlet) are given the composition\(\bar t\)tt, with theSU3 structure 3×3×3=1+8+8+10. These requirements, and the condition to minimize the number of basic particles, lead uniquely to the introduction of 6 basic particles t, one 3 triplet of charges +, 0, 0 and one\(\bar 3\) triplet of charges 0, +, +. They are associated with the values 1/3 and −1/3 of a new quantum numberZ. For the known particlesZ is equal to the baryon number. One then introduces a higher-symmetry group incorporating the new quantum number as well asSU3. This is done within the framework of simple Lie groups, and we are led to select the symplectic group in six dimensionsSp6, whose lowest dimensional representation exactly accommodates the six basic particles t. With this higher symmetry the model predicts anSU3 triplet of new pseudoscalar mesons, anSU3 sextet of new vector mesons, as well as their antiparticles, with mass separations calculable from the masses of the known mesons. For the known pseudoscalar mesons and baryons one finds the usual Gell-Mann-Okubo mass formulae, whereas for the known vector mesons one derives a mass relation identical to the formula obtained in Schwinger’sW3 symmetry model. Our model predicts in addition many new baryon resonances of high mass and large width. All mass predictions are obtained by taking a mass splitting operator of formΔm1+Δm2, whereΔm1 breaksSp6 symmetry but preservesSU3, whileΔm2 breaksSU3.Δm1 is regarded as larger thanΔm2 and is taken to second order , while forΔm2 only first-order effects are included. We are led to this assumption of a symmetry violation which is larger forSp6 than forSU3 by the circumstance that the first-order mass-splitting effects leave the new mesons to be degenerate in mass with the known ones, a situation which is unattractive in several respects.