richardson number criterion for the nonlinear stability of three dimensional stratified flow with use of a method of arnol d we derive the necessary and sufficient conditions for the formal stability of a parallel shear flow in a three dimensional stratified fluid when the local richardson number defined with respect to density variations is everywhere greater than unity the equilibrium is formally stable under nonlinear pertrubations the essential physical content of the nonlinear stability result is that the total energy acts as a potential well for deformations of the fluid across constant density surfaces this well is required to have definite curvature to assure stability under these deformations

generic bifurcation of hamiltonian systems with symmetry we study generic bifurcations of equilibria in one parameter hamiltonian systems with symmetry group on the generalized eigenvalues of the linearized system go through zero theorem 3 3 classifies expected actions of on the generalized eigenspace of this zero eigenvalue generic one degree of freedom symmetric systems is section 4 remarks concerning systems with more degrees of freedom are given in section 5

a geometric setting for internal motions of the quantum three body system quantum mechanics for internal motions of the three body system is set up on the basis of the complex vector bundle theory the three body system is called a triatomic molecule in the born oppenheimer approximation the internal states of the molecule are described as cross sections in the complex vector bundle assigned by an eigenvalue of the square of the total angular momentum operator this bundle is equipped with a linear connection which is a natural consequence of a geometric interpretation of the so called eckart condition the coupling of the internal motion with the rotation is understood naturally in terms of this connection the internal hamiltonian operator is obtained which includes the internal motion rotation coupling and a centrifugal potential the complex vector bundle for the triatomic molecule proves to be a trivial bundle though the geometric setting for the internal motion is independent of whether the bundle is trivial or not

semidirect products and reduction in mechanics this paper shows how to reduce a hamiltonian system on the cotangent bundle of a lie group to a hamiltonian system in the dual of the lie algebra of a semidirect product the procedure simplifies unifies and extends work of greene guillemin holm holmes kupershmidt marsden morrison ratiu sternberg and others the heavy top compressible fluids magnetohydrodynamics elasticity the maxwell vlasov equations and multifluid plasmas are presented as examples starting with lagrangian variables our method explains in a direct way why semidirect products occur so frequently in examples it also provides a framework for the systematic introduction of clebsch or canonical variables

canonical maps between semidirect products with applications to elasticity and superfluids it is shown that two canonical maps arising in the poisson bracket formulations of elasticity and superfluids are particular instances of general canonical maps between duals of semidirect product lie algebras

the camassa holm equation as a geodesic flow on the diffeomorphism group misiolek j geom phys 24 203 208 1998 has shown that the camassa holm equation is a geodesic flow on the bott virasoro group in this paper it is shown that the camassa holm equation for the case 0 is the geodesic spray of the weak riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating the h1 inner product over the entire group this paper uses the right trivialization technique to rigorously verify that the euler poincare theory for lie groups can be applied to diffeomorphism groups the observation made in this paper has led to physically meaningful generalizations of the ch equation to higher dimensional manifolds

a hybrid multiview stereo algorithm for modeling urban scenes we present an original multiview stereo reconstruction algorithm which allows the 3d modeling of urban scenes as a combination of meshes and geometric primitives the method provides a compact model while preserving details irregular elements such as statues and ornaments are described by meshes whereas regular structures such as columns and walls are described by primitives planes spheres cylinders cones and tori we adopt a two step strategy consisting first in segmenting the initial mesh based surface using a multilabel markov random field based model and second in sampling primitive and mesh components simultaneously on the obtained partition by a jump diffusion process the quality of a reconstruction is measured by a multi object energy model which takes into account both photo consistency and semantic considerations i e geometry and shape layout the segmentation and sampling steps are embedded into an iterative refinement procedure which provides an increasingly accurate hybrid representation experimental results on complex urban structures and large scenes are presented and compared to state of the art multiview stereo meshing algorithms

stabilization of an autonomous rolling sphere navigating in a labyrinth arena a geometric mechanics perspective abstract several concepts and results in geometric mechanics are used to analyze and control the locomotion system of an unconventional robot encapsulated in a sphere shell assumed to roll without slipping on the floor and internally equipped with a set of inertia gyros as indirect driving devices lie group symmetries intrinsic to this problem i e invariance of the system s lagrangian and velocity distribution to some group of motions allows the reduction of the equations of motion this system whose motion ability is based on angular momentum conservation is established as a controllable nonholonomic system for which the attitude position cannot be stabilized by smooth feedback laws pursuing the reduction process permits us to design a feedback law extensible to both kinematic and dynamic levels of actuation enabling the robot to execute finite time reorientation and repositioning maneuvers while confined to move in corridor like domains the derivation of the underlying nonlinearity contents via the geometric approach helps the analysis not to rely on a specific choice of coordinates and allows taking profit of the vector structure of the equations for further investigations

optimal control for holonomic and nonholonomic mechanical systems with symmetry and lagrangian reduction in this paper we establish necessary conditions for optimal control using the ideas of lagrangian reduction in the sense of reduction under a symmetry group the techniques developed here are designed for lagrangian mechanical control systems with symmetry the benefit of such an approach is that it makes use of the special structure of the system especially its symmetry structure and thus it leads rather directly to the desired conclusions for such systems lagrangian reduction can do in one step what one can alternatively do by applying the pontryagin maximum principle followed by an application of poisson reduction the idea of using lagrangian reduction in the sense of symmetry reduction was also obtained by bloch and crouch proc 33rd cdc ieee 1994 pp 2584 2590 in a somewhat different context and the general idea is closely related to those in montgomery comm math phys 128 1990 pp 565 592 and vershik and gershkovich dynamical systems vii v arnold and s p novikov eds springer verlag 1994 here we develop this idea further and apply it to some known examples such as optimal control on lie groups and principal bundles such as the ball and plate problem and reorientation examples with zero angular momentum such as the satellite with moveable masses however one of our main goals is to extend the method to the case of nonholonomic systems with a nontrivial momentum equation in the context of the work of bloch krishnaprasad marsden and murray arch rational mech anal 1996 to appear the snakeboard is used to illustrate the method

discrete time lagrangian mechanics on lie groups with an application to the lagrange top we develop the theory of discrete time lagrangian mechanics on lie groups originated in the work of veselov and moser and the theory of lagrangian reduction in the discrete time setting the results thus obtained are applied to the investigation of an integrable time discretization of a famous integrable system of classical mechanics the lagrange top we recall the derivation of the euler poinsot equations of motion both in the frame moving with the body and in the rest frame the latter ones being less widely known we find a discrete time lagrange function turning into the known continuous time lagrangian in the continuous limit and elaborate both descriptions of the resulting discrete time system namely in the body frame and in the rest frame this system naturally inherits poisson properties of the continuous time system the integrals of motion being deformed the discrete time lax representations are also found kirchhoff s kinetic analogy between elastic curves and motions of the lagrange top is also generalised to the discrete context

geometric phases reduction and lie poisson structure for the resonant three wave interaction hamiltonian lie poisson structures of the three wave equations associated with the lie algebras su 3 and su 2 1 are derived and shown to be compatible poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated these results can be applied to applications of nonlinear waves in for instance nonlinear optics some of the general structures presented in the latter part of this paper are implicit in the literature our purpose is to put the three wave interaction in the modern setting of geometric mechanics and to explore some new things such as explicit geometric phase formulas as well as some old things such as integrability in this context

robust segmentation of primitives from range data in the presence of geometric degeneracy this paper addresses a common problem in the segmentation of range images we present methods for the least squares fitting of spheres cylinders cones and tori to 3d point data and their application within a segmentation framework least squares fitting of surfaces other than planes even of simple geometric type has rarely been studied our main application areas of this research are reverse engineering of solid models from depth maps and automated 3d inspection where reliable extraction of these surfaces is essential our fitting method has the particular advantage of being robust in the presence of geometric degeneracy i e as the principal curvatures of the surfaces being fitted decrease the results returned naturally become closer and closer to those surfaces of simpler type i e planes cylinders cones or spheres which best describe the data many other methods diverge because in such cases various parameters or their combination become infinite

invariant lagrangians mechanical connections and the lagrange poincare equations we deal with lagrangian systems that are invariant under the action of a symmetry group the mechanical connection is a principal connection that is associated with lagrangians which have a kinetic energy function that is defined by a riemannian metric in this paper we extend this notion to arbitrary lagrangians we then derive the reduced lagrange poincare equations in a new fashion and we show how solutions of the euler lagrange equations can be reconstructed with the help of the mechanical connection illustrative examples confirm the theory

a game theoretic framework for landmark based image segmentation a novel game theoretic framework for landmark based image segmentation is presented landmark detection is formulated as a game in which landmarks are players landmark candidate points are strategies and likelihoods that candidate points represent landmarks are payoffs determined according to the similarity of image intensities and spatial relationships between the candidate points in the target image and their corresponding landmarks in images from the training set the solution of the formulated game theoretic problem is the equilibrium of candidate points that represent landmarks in the target image and is obtained by a novel iterative scheme that solves the segmentation problem in polynomial time the object boundaries are finally extracted by applying dynamic programming to the optimal path searching problem between the obtained adjacent landmarks the performance of the proposed framework was evaluated for segmentation of lung fields from chest radiographs and heart ventricles from cardiac magnetic resonance cross sections the comparison to other landmark based segmentation techniques shows that the results obtained by the proposed game theoretic framework are highly accurate and precise in terms of mean boundary distance and area overlap moreover the framework overcomes several shortcomings of the existing techniques such as sensitivity to initialization and convergence to local optima

on reduction of two degrees of freedom hamiltonian systems by an s1 action and so0 1 2 as a dynamical group reduction by an s1 action is a method of finding periodic solutions in hamiltonian systems which is known rather as the method of averaging such periodic solutions can be reconstructed as s1 orbits by pulling back the critical points of an associated reduced hamiltonian on a reduced phase space along the reduction for hamiltonian systems of two degrees of freedom a geometric setting of the reduction is already accomplished in the case where the reduced phase space is a two sphere in the euclidean space r3 and the reduced hamilton s equations of motion are euler s equations this article deals with the case where the reduced phase space will be a two hyperboloid in the three minkowski space and the reduced hamilton s equations of motion will be euler s equations with respect to the lorentz metric this reduction is associated with su 1 1 symplectic action on the phase space r4 as a consequence of this association the reduced hamiltonian system proves to admit a dynamical group so0 1 2 a w

coadjoint orbits vortices and clebsch variables for incompressible fluids this paper is a study of incompressible fluids especially their clebsch variables and vortices using symplectic geometry and the lie poisson structure on the dual of a lie algebra following ideas of arnold and others it is shown that euler s equations are lie poisson equations associated to the group of volume preserving diffeomorphisms the dual of the lie algebra is seen to be the space of vortices and kelvin s circulation theorem is interpreted as preservation of coadjoint orbits in this context clebsch variables can be understood as momentum maps the motion of n point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit and the standard canonical variables for them are a special kind of clebsch variables point vortices with cores vortex patches and vortex filaments can be understood in a similar way this leads to an explanation of the geometry behind the hald beale majda convergence theorems for vorticity algorithms symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the kdv and the schrodinger equations respectively

symmetry and bifurcations of momentum mappings the zero set of a momentum mapping is shown to have a singularity at each point with symmetry the zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function the proof uses the kuranishi theory of deformations among the applications it is shown that the set of all solutions of the yang mills equations on a lorentz manifold has a singularity at any solution with symmetry in the sense of a pure gauge symmetry similarly the set of solutions of einstein s equations has a singularity at any solution that has spacelike killing fields provided the spacetime has a compact cauchy surface

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

the symmetry group of the harmonic oscillator and its reduction the symmetry of the harmonic oscillator is dealt with in the hamiltonian formalism unitary operators representing the symmetry are studied from this point of view of additional interest is reduction of the symmetry group su 4 for the four dimensional harmonic oscillator subspaces are determined from the representation spaces for su 4 so as to give those for so 4

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

range image segmentation by an effective jump diffusion method this paper presents an effective jump diffusion method for segmenting a range image and its associated reflectance image in the bayesian framework the algorithm works on complex real world scenes indoor and outdoor which consist of an unknown number of objects or surfaces of various sizes and types such as planes conics smooth surfaces and cluttered objects like trees and bushes formulated in the bayesian framework the posterior probability is distributed over a solution space with a countable number of subspaces of varying dimensions the algorithm simulates markov chains with both reversible jumps and stochastic diffusions to traverse the solution space the reversible jumps realize the moves between subspaces of different dimensions such as switching surface models and changing the number of objects the stochastic langevin equation realizes diffusions within each subspace to achieve effective computation the algorithm precomputes some importance proposal probabilities over multiple scales through hough transforms edge detection and data clustering the latter are used by the markov chains for fast mixing the algorithm is tested on 100 1d simulated data sets for performance analysis on both accuracy and speed then the algorithm is applied to three data sets of range images under the same parameter setting the results are satisfactory in comparison with manual segmentations

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

variational integrators and the newmark algorithm for conservative and dissipative mechanical systems the purpose of this work is twofold first we demonstrate analytically that the classical newmark family as well as related integration algorithms are variational in the sense of the veselov formulation of discrete mechanics such variational algorithms are well known to be symplectic and momentum preserving and to often have excellent global energy behavior this analytical result is veried through numerical examples and is believed to be one of the primary reasons that this class of algorithms performs so well second we develop algorithms for mechanical systems with forcing and in particular for dissipative systems in this case we develop integrators that are based on a discretization of the lagrange d alembert principle as well as on a variational formulation of dissipation it is demonstrated that these types of structured integrators have good numerical behavior in terms of obtaining the correct amounts by which the energy changes over the integration run

the hamiltonian structure of the maxwell vlasov equations morrison 25 has observed that the maxwell vlasov and poisson vlasov equations for a collisionless plasma can be written in hamiltonian form relative to a certain poisson bracket we derive another poisson structure for these equations by using general methods of symplectic geometry the main ingredients in our construction are the symplectic structure on the co adjoint orbits for the group of canonical transformations and the symplectic structure for the phase space of the electromagnetic field regarded as a gauge theory our poisson bracket satisfies the jacobi identity whereas morrison s does not 37 our construction also shows where canonical variables can be found and can be applied to the yang mills vlasov equations and to electromagnetic fluid dynamics

heckman kostant and steinberg formulas for symplectic manifolds abstract let m be a compact symplectic manifold on which a compact connected lie group k acts in a hamiltonian fashion in part i we derive a formula for j m which generalizes the heckman formula for co adjoint orbits here j is the moment map associated with the action of k on m and m the liouville measure on m in part ii we derive a quantum analogue of this formula which extends to the symplectic setting classical multiplicity formulas of kostant and steinberg

feature selection in the pattern classification problem of digital chest radiograph segmentation in pattern classification problems the choice of variables to include in the feature vector is a difficult one the authors have investigated the use of stepwise discriminant analysis as a feature selection step in the problem of segmenting digital chest radiographs in this problem locally calculated features are used to classify pixels into one of several anatomic classes the feature selection step was used to choose a subset of features which gave performance equivalent to the entire set of candidate features while utilizing less computational resources the impact of using the reduced selected feature set on classifier performance is evaluated for two classifiers a linear discriminator and a neural network the results from the reduced selected feature set were compared to that of the full feature set as well as a randomly selected reduced feature set the results of the different feature sets were also compared after applying an additional postprocessing step which used a rule based spatial information heuristic to improve the classification results this work shows that in the authors pattern classification problem using a feature selection step reduced the number of features used reduced the processing time requirements and gave results comparable to the full set of features

stereo processing by semiglobal matching and mutual information this paper describes the semiglobal matching sgm stereo method it uses a pixelwise mutual information ml based matching cost for compensating radiometric differences of input images pixelwise matching is supported by a smoothness constraint that is usually expressed as a global cost function sgm performs a fast approximation by pathwise optimizations from all directions the discussion also addresses occlusion detection subpixel refinement and multibaseline matching additionally postprocessing steps for removing outliers recovering from specific problems of structured environments and the interpolation of gaps are presented finally strategies for processing almost arbitrarily large images and fusion of disparity images using orthographic projection are proposed a comparison on standard stereo images shows that sgm is among the currently top ranked algorithms and is best if subpixel accuracy is considered the complexity is linear to the number of pixels and disparity range which results in a runtime of just 1 2 seconds on typical test images an in depth evaluation of the ml based matching cost demonstrates a tolerance against a wide range of radiometric transformations finally examples of reconstructions from huge aerial frame and pushbroom images demonstrate that the presented ideas are working well on practical problems

routh reduction and the class of magnetic lagrangian systems in this paper some new aspects related to routh reduction of lagrangian systems with symmetry are discussed the main result of this paper is the introduction of a new concept of transformation that is applicable to systems obtained after routh reduction of lagrangian systems with symmetry so called magnetic lagrangian systems we use these transformations in order to show that under suitable conditions the reduction with respect to a full semi direct product group is equivalent to the reduction with respect to an abelian normal subgroup the results in this paper are closely related to the more general theory of routh reduction by stages

isoholonomic problems and some applications we study the problem of finding the shortest loops with a given holonomy we show that the solutions are the trajectories of particles in yang mills potentials theorem 4 or equivalently the projections of kaluza klein geodesics theorem 2 applications to quantum mechanics berry s phase sect 3 and the optimal control of deformable bodies sect 6 are touched upon

the maxwell vlasov equations in euler poincare form low s well known action principle for the maxwell vlasov equations of ideal plasma dynamics was originally expressed in terms of a mixture of eulerian and lagrangian variables by imposing suitable constraints on the variations and analyzing invariance properties of the lagrangian as one does for the euler equations for the rigid body and ideal fluids we first transform this action principle into purely eulerian variables hamilton s principle for the eulerian description of low s action principle then casts the maxwell vlasov equations into euler poincare form for right invariant motion on the diffeomorphism group of position velocity phase space openface r 6 legendre transforming the eulerian form of low s action principle produces the hamiltonian formulation of these equations in the eulerian description since it arises from euler poincare equations this hamiltonian formulation can be written in terms of a poisson structure that contains the lie poisson bracket on the dual of a semidirect product lie algebra because of degeneracies in the lagrangian the legendre transform is dealt with using the dirac theory of constraints another maxwell vlasov poisson structure is known whose ingredients are the lie poisson bracket on the dual of the lie algebra of symplectomorphisms of phase space and the born infeld brackets for the maxwell field we discuss the relationship between these two hamiltonian formulations we also discuss the general kelvin noether theorem for euler poincare equations and its meaning in the plasma context

fast approximate energy minimization via graph cuts many tasks in computer vision involve assigning a label such as disparity to every pixel a common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist e g at object boundaries these tasks are naturally stated in terms of energy minimization the authors consider a wide class of energies with various smoothness constraints global minimization of these energy functions is np hard even in the simplest discontinuity preserving case therefore our focus is on efficient approximation algorithms we present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves namely expansion moves and swap moves these moves can simultaneously change the labels of arbitrarily large sets of pixels in contrast many standard algorithms including simulated annealing use small moves where only one pixel changes its label at a time our expansion algorithm finds a labeling within a known factor of the global minimum while our swap algorithm handles more general energy functions both of these algorithms allow important cases of discontinuity preserving energies we experimentally demonstrate the effectiveness of our approach for image restoration stereo and motion on real data with ground truth we achieve 98 percent accuracy

routh s procedure for non abelian symmetry groups we extend routh s reduction procedure to an arbitrary lagrangian system that is one whose lagrangian is not necessarily the difference of kinetic and potential energies with a symmetry group which is not necessarily abelian to do so we analyze the restriction of the euler lagrange field to a level set of momentum in velocity phase space we present a new method of analysis based on the use of quasivelocities we discuss the reconstruction of solutions of the full euler lagrange equations from those of the reduced equations

hierarchical lung field segmentation with joint shape and appearance sparse learning lung field segmentation in the posterior anterior pa chest radiograph is important for pulmonary disease diagnosis and hemodialysis treatment due to high shape variation and boundary ambiguity accurate lung field segmentation from chest radiograph is still a challenging task to tackle these challenges we propose a joint shape and appearance sparse learning method for robust and accurate lung field segmentation the main contributions of this paper are 1 a robust shape initialization method is designed to achieve an initial shape that is close to the lung boundary under segmentation 2 a set of local sparse shape composition models are built based on local lung shape segments to overcome the high shape variations 3 a set of local appearance models are similarly adopted by using sparse representation to capture the appearance characteristics in local lung boundary segments thus effectively dealing with the lung boundary ambiguity 4 a hierarchical deformable segmentation framework is proposed to integrate the scale dependent shape and appearance information together for robust and accurate segmentation our method is evaluated on 247 pa chest radiographs in a public dataset the experimental results show that the proposed local shape and appearance models outperform the conventional shape and appearance models compared with most of the state of the art lung field segmentation methods under comparison our method also shows a higher accuracy which is comparable to the inter observer annotation variation

insertion of 3 d primitives in mesh based representations towards compact models preserving the details we propose an original hybrid modeling process of urban scenes that represents 3 d models as a combination of mesh based surfaces and geometric 3 d primitives meshes describe details such as ornaments and statues whereas 3 d primitives code for regular shapes such as walls and columns starting from an 3 d surface obtained by multiview stereo techniques these primitives are inserted into the surface after being detected this strategy allows the introduction of semantic knowledge the simplification of the modeling and even correction of errors generated by the acquisition process we design a hierarchical approach exploring different scales of an observed scene each level consists first in segmenting the surface using a multilabel energy model optimized by expansion and then in fitting 3 d primitives such as planes cylinders or tori on the obtained partition where relevant experiments on real meshes depth maps and synthetic surfaces show good potential for the proposed approach

complex form reduction and lie poisson structure for the nonlinearized spectral problem of the heisenberg hierarchy abstract in this paper the relation between the different restricted systems associated with the heisenberg magnetic hm equation is studied by the reduction procedure with the help of a lie group homomorphism of su 2 into so 3 the euler rodriguez type parameters are introduced to generate new finite dimensional integrable system it has shown that the resulting system which is the nonlinearized spectral problem of hm hierarchy on c2n is a hamiltonian system in complex form further poisson reduction and lie poisson structure are derived by the method of invariants the reduced system is found to be a hamiltonian system on the orbit space c2n tn r3n coinciding with the nonlinearized lenard spectral problem moreover the fully reduced systems on the leaves of the symplectic foliation are also given specifically the reduction extended to the common level set of the complex cones is the usual 2 2 nonlinearized spectral problem finally the integrability of the system with lie poisson structure is proven by making use of the so 3 symmetry

controlled lagrangians and the stabilization of mechanical systems i the first matching theorem we develop a method for the stabilization of mechanical systems with symmetry based on the technique of controlled lagrangians the procedure involves making structured modifications to the lagrangian for the uncontrolled system thereby constructing the controlled lagrangian the euler lagrange equations derived from the controlled lagrangian describe the closed loop system where new terms in these equations are identified with control forces since the controlled system is lagrangian by construction energy methods can be used to find control gains that yield closed loop stability we use kinetic shaping to preserve symmetry and only stabilize systems module the symmetry group the procedure is demonstrated for several underactuated balance problems including the stabilization of an inverted planar pendulum on a cart moving on a line and an inverted spherical pendulum on a cart moving in the plane

the connection whose holonomy is the classical adiabatic angles of hannay and berry and its generalization to the non integrable case we show how averaging defines an ehresmann connection whose holonomy is the classical adiabatic angles which hannay defined for families of completely integrable systems the averaging formula we obtain for the connection only requires that the family of hamiltonians has a continuous symmetry group this allows us to extend the notion of the hannay angles to families of non integrable systems with symmetry we state three geometric axioms satisfied by the connection these axioms uniquely determine the connection thus enabling us to find new formulas for the connection and its curvature two examples are given

reduction of exterior differential systems for ordinary variational problems this paper deals with the reduction of ordinary variational problems with abelian and non abelian symmetry groups it differs from the existing results on geometric reduction in the approach that is completely lagrangian and based on exterior differential systems in this setting two procedures of reduction come out naturally the first one is the lagrangian analog of the marsden weinstein reduction holding in the symplectic case and the other one especially in the non abelian case comes out naturally in our setting

variational principles surface evolution pdes level set methods and the stereo problem we present a novel geometric approach for solving the stereo problem for an arbitrary number of images spl ges 2 it is based upon the definition of a variational principle that must be satisfied by the surfaces of the objects in the scene and their images the euler lagrange equations that are deduced from the variational principle provide a set of partial differential equations pdes that are used to deform an initial set of surfaces which then move toward the objects to be detected the level set implementation of these pdes potentially provides an efficient and robust way of achieving the surface evolution and to deal automatically with changes in the surface topology during the deformations i e to deal with multiple objects results of an implementation of our theory also dealing with occlusion and visibility are presented on synthetic and real images

reduction theory and the lagrange routh equations reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of euler jacobi lagrange hamilton routh poincare and others the modern vision of mechanics includes besides the traditional mechanics of particles and rigid bodies field theories such as electromagnetism fluid mechanics plasma physics solid mechanics as well as quantum mechanics and relativistic theories including gravity symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas to subtle symmetries underlying integrable systems reduction theory concerns the removal of symmetries and their associated conservation laws variational principles along with symplectic and poisson geometry provide fundamental tools for this endeavor reduction theory has been extremely useful in a wide variety of areas from a deeper understanding of many physical theories including new variational and poisson structures to stability theory integrable systems as well as geometric phases this paper surveys progress in selected topics in reduction theory especially those of the last few decades as well as presenting new results on non abelian routh reduction we develop the geometry of the associated lagrange routh equations in some detail the paper puts the new results in the general context of reduction theory and discusses some future directions

symplectic geometry and topology this is a review of the ideas underlying the application of symplectic geometry to hamiltonian systems the paper begins with symplectic manifolds and their lagrangian submanifolds covers contact manifolds and their legendrian submanifolds and indicates the first steps of symplectic and contact topology

the riemannian structure of euclidean shape spaces a novel environment for statistics the riemannian metric structure of the shape space sum k m for k labelled points in mathbb r m was given by kendall for the atypically simple situations in which m 1 or 2 and k geq 2 here we deal with the general case m geq 1 k geq 2 by using the properties of riemannian submersions and warped products as studied by o neill the approach is via the associated size and shape space that is the warped product of the shape space and the half line mathbb r carrying size the warping function being equal to the square of the size when combined with parallel studies by le of the corresponding global geodesic geometry the results obtained here determine the environment in which shape statistical calculations have to be acted out finally three different applications are discussed that illustrate the theory and its use in practice

reduced dynamics and lagrangian submanifolds of symplectic manifolds in this paper we will see that the symplectic creed by weinstein everything is a lagrangian submanifold also holds for hamilton poincare and lagrange poincare reduction in fact we show that solutions of the hamilton poincare equations and of the lagrange poincare equations are in one to one correspondence with distinguished curves in a lagrangian submanifold of a symplectic manifold for this purpose we will combine the concept of a tulczyjew triple with marsden weinstein symplectic reduction

lin constraints clebsch potentials and variational principles the poisson bracket formulation of fluid plasma and rigid body type systems has undergone considerable recent development using techniques of symmetry group reduction the relationship between this approach and that using lin constraints and clebsch potentials is established the connection is made in the setting of abstract clebsch variables as well as that of variational principles on reduced spaces variational principles for both the clebsch and reduced form such as fluids in spatial representation are derived from the standard variational principle of hamilton in material lagrangian representation using reduction theory

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

discrete routh reduction this paper develops the theory of abelian routh reduction for discrete mechanical systems and applies it to the variational integration of mechanical systems with abelian symmetry the reduction of variational runge kutta discretizations is considered as well as the extent to which symmetry reduction and discretization commute these reduced methods allow the direct simulation of dynamical features such as relative equilibria and relative periodic orbits that can be obscured or difficult to identify in the unreduced dynamics the methods are demonstrated for the dynamics of an earth orbiting satellite with a non spherical j2 correction as well as the double spherical pendulum the j2 problem is interesting because in the unreduced picture geometric phases inherent in the model and those due to numerical discretization can be hard to distinguish but this issue does not appear in the reduced algorithm where one can directly observe interesting dynamical structures in the reduced phase space the cotangent bundle of shape space in which the geometric phases have been removed the main feature of the double spherical pendulum example is that it has a non trivial magnetic term in its reduced symplectic form our method is still efficient as it can directly handle the essential non canonical nature of the symplectic structure in contrast a traditional symplectic method for canonical systems could require repeated coordinate changes if one is evoking darboux theorem to transform the symplectic structure into canonical form thereby incurring additional computational cost our method allows one to design reduced symplectic integrators in a natural way despite the non canonical nature of the symplectic structure

poisson brackets and clebsch representations for magnetohydrodynamics multifluid plasmas and elasticity abstract poisson brackets are constructed by the same mathematical procedure for three physical theories ideal magnetohydrodynamics multifluid plasmas and elasticity each of these brackets is given a simple lie algebraic interpretation moreover each bracket is induced to physical space by use of a canonical poisson bracket in the space of clebsch potentials which are constructed for each physical theory by the standard procedure of constrained lagrangians

nonholonomic mechanical systems with symmetry this work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of lagrangian mechanics and with a view to control theoretical applications the basic methodology is that of geometric mechanics applied to the formulation of lagrange d alembert generalizing the use of connections and momentum maps associated with a given symmetry group to this case we begin by formulating the mechanics of nonholonomic systems using an ehresmann connection to model the constraints and show how the curvature of this connection enters into lagrange s equations unlike the situation with standard configuration space constraints the presence of symmetries in the nonholonomic case may or may not lead to conservation laws however the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables this momentum equation which plays an important role in control problems involves parallel transport operators and is computed explicitly in coordinates an alternative description using a body reference frame relates part of the momentum equation to the components of the euler poincar e equations along those symmetry directions consistent with the constraints one of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not an example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard a modified version of the skateboard which uses momentum coupling for locomotion generation we construct a synthesis of the mechanical connection and the ehresmann connection defining the constraints obtaining an important new object we call the nonholonomic connection when the nonholonomic connection is a principal connection for the given symmetry group we show how to perform lagrangian reduction in the presence of nonholonomic constraints generalizing previous results which only held in special cases several detailed examples are given to illustrate the theory september 1994 revised march 1995 revised june 1995

stratified symplectic spaces and reduction let m w be a hamiltonian g space with proper momentum map j m g it is well known that if zero is a regular value of j and g acts freely on the level set j 0 then the reduced space mo j 0 g is a symplectic manifold we show that if the regularity assumptions are dropped the space mo is a union of symplectic manifolds i e it is a stratified symplectic space arms et al 2 proved that mo possesses a natural poisson bracket using their result we study hamiltonian dynamics on the reduced space in particular we show that hamiltonian flows are strata preserving and give a recipe for lifting a reduced hamiltonian flow to the level set j 0 finally we give a detailed description of the stratification of mo and prove the existence of a connected

cosmos a representation scheme for 3d free form objects we address the problem of representing and recognizing 3d free form objects when 1 the object viewpoint is arbitrary 2 the objects may vary in shape and complexity and 3 no restrictive assumptions are made about the types of surfaces on the object we assume that a range image of a scene is available containing a view of a rigid 3d object without occlusion we propose a new and general surface representation scheme for recognizing objects with free form sculpted surfaces in this scheme an object is described concisely in terms of maximal surface patches of constant shape index the maximal patches that represent the object are mapped onto the unit sphere via their orientations and aggregated via shape spectral functions properties such as surface area curvedness and connectivity which are required to capture local and global information are also built into the representation the scheme yields a meaningful and rich description useful for object recognition a novel concept the shape spectrum of an object is also introduced within the framework of cosmos for object view grouping and matching we demonstrate the generality and the effectiveness of our scheme using real range images of complex objects

stabilization of relative equilibria discusses the problem of obtaining feedback laws to asymptotically stabilize relative equilibria of mechanical systems with symmetry we show how to stabilize an internally unstable relative equilibrium using internal actuators the methodology is that of potential shaping but the system is allowed to be underactuated i e have fewer actuators than the dimension of the shape space the theory is illustrated with the problem of stabilization of the cowboy relative equilibrium of the double spherical pendulum

mechanical integrators derived from a discrete variational principle many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion in this paper we present a procedure to construct time stepping algorithms that approximate the flow of continuous odes for mechanical systems by discretizing hamilton s principle rather than the equations of motion the discrete equations share similarities to the continuous equations by preserving invariants including the symplectic form and the momentum map we first present a formulation of discrete mechanics along with a discrete variational principle we then show that the resulting equations of motion preserve the symplectic form and that this formulation of mechanics leads to conservation laws from a discrete version of noether s theorem we then use the discrete mechanics formulation to develop a procedure for constructing symplectic momentum mechanical integrators for lagrangian systems with holonomic constraints we apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators

the kowalewski top 99 years later a lax pair generalizations and explicit solutions a natural lax pair for the kowalewski top is derived by using a general group theoretic approach this gives a new insight into the algebraic geometry of the top and leads to its complete solution via finite band integration theory

local equilibrium controllability of multibody systems controlled via shape change we study local equilibrium controllability of shape controlled multibody systems the multibody systems are defined on a trivial principal fiber bundle by a lagrangian that characterizes the base body motion and shape dynamics a potential dependent on an advected parameter e g uniform gravitational potential is considered this potential breaks base body symmetries but a symmetry subgroup is assumed to exist symmetric product formulas are derived and important properties are obtained for symmetric products of horizontal shape control vector fields and a potential vector field that is dependent on an advected parameter based on these properties sufficient conditions for local equilibrium controllability and local fiber equilibrium controllability are developed these results are applied to two classes of shape controlled multibody systems in a uniform gravitational field multibody attitude systems and neutrally buoyant multibody underwater vehicles

reconstruction equations and the karhunen loeve expansion for systems with symmetry we present a method for applying the karhunen loeve decomposition to systems with continuous symmetry the techniques in this paper contribute to the general procedure of removing variables associated with the symmetry of a problem and related ideas have been used in previous works both to identify coherent structures in solutions of pdes and to derive low order models via galerkin projection the main result of this paper is to derive a simple and easily implementable set of reconstruction equationswhich close the system of odes produced by galerkin projection the geometric interpretation of the method closely parallels techniques used in geometric phases and reconstruction techniques in geometric mechanics we apply the method to the kuramoto sivashinsky equation and are able to derive accurate models of considerably lower dimension than are possible with the traditional karhunen loeve expansion