on computing the distance to stability for matrices using linear dissipative hamiltonian systems abstract in this paper we consider the problem of computing the nearest stable matrix to an unstable one we propose new algorithms to solve this problem based on a reformulation using linear dissipative hamiltonian systems we show that a matrix a is stable if and only if it can be written as a j r q where j j t r 0 and q 0 that is r is positive semidefinite and q is positive definite this reformulation results in an equivalent optimization problem with a simple convex feasible set we propose three strategies to solve the problem in variables j r q i a block coordinate descent method ii a projected gradient descent method and iii a fast gradient method inspired from smooth convex optimization these methods require o n 3 operations per iteration where n is the size of a we show the effectiveness of the fast gradient method compared to the other approaches and to several state of the art algorithms

on the k tuple domination of generalized de brujin and kautz digraphs a vertex u in a digraph g v a is said to dominate itself and vertices v such that u v a for a positive integer k a k tuple dominating set of g is a subset d of vertices such that every vertex in g is dominated by at least k vertices in d the k tuple domination number of g is the minimum cardinality of a k tuple dominating set of g this paper deals with the k tuple domination problem on generalized de bruijn and kautz digraphs we establish bounds on the k tuple domination number for the generalized de bruijn and kautz digraphs and we obtain some conditions for the k tuple domination number attaining the bounds

geometric description of vakonomic and nonholonomic dynamics comparison of solutions we treat the vakonomic dynamics with general constraints within a new geometric framework which can be useful in the study of optimal control problems we compare our formulation with the one of vershik and gershkovich in the case of linear constraints we show how nonholonomic mechanics also admits a new geometrical description which allows us to develop an algorithm of comparison between the solutions of both dynamics examples illustrating the theory are treated

nonrigid coregistration of diffusion tensor images using a viscous fluid model and mutual information in this paper a nonrigid coregistration algorithm based on a viscous fluid model is proposed that has been optimized for diffusion tensor images dti in which image correspondence is measured by the mutual information criterion several coregistration strategies are introduced and evaluated both on simulated data and on brain intersubject dti data two tensor reorientation methods have been incorporated and quantitatively evaluated simulation as well as experimental results show that the proposed viscous fluid model can provide a high coregistration accuracy although the tensor reorientation was observed to be highly sensitive to the local deformation field nevertheless this coregistration method has demonstrated to significantly improve spatial alignment compared to affine image matching

euler lagrange and hamilton equations for non holonomic systems in field theory a generalization of the concept of a system of non holonomic constraints to fibred manifolds with n dimensional bases is considered motion equations in both lagrangian and hamiltonian settings for systems subjected to such constraints are investigated regularity conditions for the existence of a non holonomic legendre transformation and the corresponding formulae for hamiltonian and momenta are found in particular lagrangian constraints and semi holonomic constraints and simplifications arising in this case are discussed

geometric hamilton jacobi theory for higher order autonomous systems the geometric framework for the hamilton jacobi theory is used to study this theory in the ambient of higher order mechanical systems both in the lagrangian and hamiltonian formalisms thus we state the corresponding hamilton jacobi equations in these formalisms and apply our results to analyze some particular physical examples

regularity and symmetries of nonholonomic systems lagrangian systems with nonholonomic constraints may be considered as singular differential equations defined by some constraints and some multipliers the geometry solutions symmetries and constants of motion of such equations are studied within the framework of linearly singular differential equations some examples are given in particular the well known singular lagrangian of the relativistic particle which with the nonholonomic constraint v2 c2 yields a regular system

hamilton jacobi theory in multisymplectic classical field theories the geometric framework for the hamilton jacobi theory developed in the studies of carinena et al int j geom methods mod phys 3 7 1417 1458 2006 carinena et al int j geom methods mod phys 13 2 1650017 2015 and de leon et al variations geometry and physics nova science publishers new york 2009 is extended for multisymplectic first order classical field theories the hamilton jacobi problem is stated for the lagrangian and the hamiltonian formalisms of these theories as a particular case of a more general problem and the classical hamilton jacobi equation for field theories is recovered from this geometrical setting particular and complete solutions to these problems are defined and characterized in several equivalent ways in both formalisms and the equivalence between them is proved the use of distributions in jet bundles that represent the solutions to the field equations is the fundamental tool in this formulation some examples are analyzed and in particular the h

reduction and reconstruction of the dynamics of nonholonomic systems the reduction and reconstruction of the dynamics of nonholonomic mechanical systems with symmetry are investigated we have considered a more general framework of constrained hamiltonian systems since they appear in the reduction procedure a reduction scheme in terms of the nonholonomic momentum mapping is developed the reduction of the nonholonomic brackets is also discussed the theory is illustrated with several examples

mechanical systems subjected to generalized non holonomic constraints we study mechanical systems subject to constraint functions that can be dependent at some points and independent at the rest such systems are modelled by means of generalized codistributions we discuss how the constraint force can transmit an impulse to the motion at the points of dependence and derive an explicit formula to obtain the post impact momentum in terms of the pre impact momentum

a class of second order difference approximations for solving space fractional diffusion equations a class of second order approximations called the weighted and shifted grunwald difference wsgd operators are proposed for riemann liouville fractional derivatives with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions the stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established several numerical examples are implemented to test the efficiency of the numerical schemes and confirm the convergence order and the numerical results for variable coefficients problem are also presented

an alternating direction implicit scheme of a fractional order diffusion tensor image registration model abstract in this paper we propose an alternating direction implicit adi scheme of a fractional order diffusion tensor image dti registration model furthermore compatibility stability and convergence of adi scheme are proved moreover five numerical tests are also performed to show the efficiency of this scheme

lagrangian systems with constraints a geometric approach to the method of lagrange multipliers a geometric approach to the method of lagrange multipliers is presented using the framework of the tangent bundle geometry the nonholonomic systems with constraint functions linear in the velocities are studied in the first place and then and using this study of the nonholonomic mechanical systems as a previous result the holonomic systems are considered the lagrangian inverse problem is also analysed and finally the theory is illustrated with several examples

on the hamilton jacobi theory for singular lagrangian systems we develop a hamilton jacobi theory for singular lagrangian systems using the gotay nester hinds constraint algorithm the procedure works even if the system has secondary constraints

a unified approach to fast image registration and a new curvature based registration technique abstract image registration is central to many challenges in medical imaging today it has a vast range of applications the purpose of this note is twofold first we review some of the most promising non linear registration strategies currently used in medical imaging we show that all these techniques may be phrased in terms of a variational problem and allow for a unified treatment second we introduce within the variational framework a new non linear registration model based on a curvature type smoother we show that affine linear transformations belong to the kernel of this regularizer as a result the approach becomes more robust against poor initializations of a pre registration step furthermore we develop a stable and fast implementation of the new scheme based on a real discrete cosine transformation we demonstrate the advantages of the new technique for synthetic data sets and present an application of the algorithm for registering mr mammography images

variational image registration by a total fractional order variation model in this paper a new framework of nonlocal deformation in non rigid image registration is presented it is well known that many non rigid image registration techniques may lead to unsteady deformation e g not one to one if the dissimilarity between the reference and template images is too large we present a novel variational framework of the total fractional order variation to derive the underlying fractional euler lagrange equations and a numerical implementation combining the semi implicit update and conjugate gradients cg solution to solve the nonlinear systems numerical experiments show that the new registration not only produces accurate and smooth solutions but also allows for a large rigid alignment the evaluations of the new model demonstrate substantial improvements in accuracy and robustness over the conventional image registration approaches

a geometrical framework for the study of non holonomic lagrangian systems a geometrical framework is presented for the treatment of a class of dynamical systems which are modelled by a system of second order differential equations coupled with first order equations linear in the derivatives such problems in particular make their appearance in the study of lagrangian systems with non holonomic constraints among other things we discuss the concepts of symmetry and adjoint symmetry for such systems and identify for that purpose an appropriate notion of dynamical covariant derivative and jacobi endomorphism the intrinsic tools which are being developed further allow a direct geometrical construction of the dynamics of non holonomic systems the vertically rolling disc is chosen as an illustrative example for the newly proposed formalism and results

time dependent lagrangian systems a geometric approach to the theory of systems with constraints a geometric approach to the theory of time dependent regular lagrangian systems with constraints is presented using the framework of the exact contact manifold tq r l the main subject of the article concerns the properties of the time dependent nonholonomic constraints and the geometric approach to the method of the lagrange multipliers it is shown that every constraint determines a contact one form a vertical vector field and a nonvertical vector field the explicit form of the vector field representing the constrained dynamics is obtained and finally the properties of all these one forms and vector fields are discussed

approximate dissipative hamiltonian realization and construction of local lyapunov functions the key in applying energy based control approach is to be able to express the system under consideration as a dissipative hamiltonian system i e to obtain dissipative hamiltonian realization dhr for the system in general the precise dhr form is hard to obtain for nonlinear dynamic systems when a precise dhr does not exist for a dynamic system or such a precise realization is difficulty to obtain it is necessary to consider its approximate realization this paper investigates approximate dhr and construction of local lyapunov functions for time invariant nonlinear systems it is shown that every nonlinear affine system has an approximate dhr if linearization of the system is controllable based on the diagonal normal form of nonlinear dynamic systems a new algorithm is established for the approximate dhr finally we present the concept of kth degree approximate lyapunov function and provide a method to construct such a lyapunov function example studies show that the methodology presented in this paper is very effective

on the geometry of non holonomic lagrangian systems we present a geometric framework for non holonomic lagrangian systems in terms of distributions on the configuration manifold if the constrained system is regular an almost product structure on the phase space of velocities is constructed such that the constrained dynamics is obtained by projecting the free dynamics if the constrained system is singular we develop a constraint algorithm which is very similar to that developed by dirac and bergmann and later globalized by gotay and nester special attention to the case of constrained systems given by connections is paid in particular we extend the results of koiller for caplygin systems an application to the so called non holonomic geometry is given

a hamilton jacobi theory for implicit differential systems in this paper we propose a geometric hamilton jacobi theory for systems of implicit differential equations in particular we are interested in implicit hamiltonian systems described in terms of lagrangian submanifolds of tt q generated by morse families the implicit character implies the nonexistence of a hamiltonian function describing the dynamics this fact is here amended by a generating family of morse functions which plays the role of a hamiltonian a hamilton jacobi equation is obtained with the aid of this generating family of functions to conclude we apply our results to singular lagrangians by employing the construction of special symplectic structures

finding minimum weight connected dominating set in stochastic graph based on learning automata finding the minimum weight connected dominating set mcds in an arbitrary graph is an np hard problem and several heuristics and approximation methods have been proposed to solve it forwarding the messages along the virtual backbone induced by the connected dominating set cds significantly reduces the routing overhead as well as the power consumption by reducing the routing nodes to the backbone nodes this paper first defines the stochastic mcds problem where the probability distribution function pdf of the random weight associated with the graph vertices is unknown then it presents several learning automata based algorithms algorithms 1 6 to solve the stochastic mcds problem taking advantage of learning automata the proposed algorithms significantly reduce the number of samples that must be taken from the graph to construct the mcds it is proved that by the proper choice of the learning rate the probability of finding the mcds is close enough to unity the standard sampling method ssm is the baseline with which we compare the performance of the proposed algorithms experimental results show that algorithm 6 significantly outperforms the ssm and the other proposed algorithms in terms of the sampling rate

the nonholonomic variational principle a variational principle for mechanical systems and fields subject to nonholonomic constraints is found providing chetaev reduced equations as equations for extremals investigating nonholonomic variations of the chetaev type and their properties we develop foundations of the calculus of variations on constraint manifolds modelled as fibred submanifolds in jet bundles this setting is appropriate to study general first order nonlinear nonitegrable constraints that locally are given by a system of first order ordinary or partial differential equations we obtain an invariant constrained first variation formula and constrained euler lagrange equations both in intrinsic and coordinate forms and show that the equations are the same as chetaev equations without lagrange multipliers introduced recently by other methods we pay attention to two possible settings first when the constrained system arises from an unconstrained lagrangian system defined in a neighbourhood of the constraint and second more generally when an internal constrained system on the constraint manifold is given in the latter case a corresponding unconstrained system need not be a lagrangian nor even exist we also study in detail an important particular case nonholonomic constraints that can be alternatively modelled by means of co distributions in the total space of the fibred manifold in nonholonomic mechanics this happens whenever constraints affine in velocities are considered it becomes clear that and why if the distribution is completely integrable the constraints are semiholonomic the principle of virtual displacements holds and can be used to obtain the constrained first variational formula by a more or less standard procedure traditionally used when unconstrained or holonomic systems are concerned if however the constraint is nonintegrable no significant simplifications are available among others some properties of nonholonomic systems are clarified that without a deeper insight seem rather mysterious

computing large deformation metric mappings via geodesic flows of diffeomorphisms this paper examine the euler lagrange equations for the solution of the large deformation diffeomorphic metric mapping problem studied in dupuis et al 1998 and trouve 1995 in which two images i 0 i 1 are given and connected via the diffeomorphic change of coordinates i 0 1 i 1 where 1 is the end point at t 1 of curve t t 0 1 satisfying t v t t t 0 1 with 0 id the variational problem takes the form mathop arg text m in limits upsilon dot phi t upsilon t left dot phi right left int 0 1 left upsilon t right 2 text d t left i 0 circ phi 1 1 i 1 right l 2 2 right where v t v is an appropriate sobolev norm on the velocity field v t and the second term enforces matching of the images with l 2 representing the squared error norm in this paper we derive the euler lagrange equations characterizing the minimizing vector fields v t t 0 1 assuming sufficient smoothness of the norm to guarantee existence of solutions in the space of diffeomorphisms we describe the implementation of the euler equations using semi lagrangian method of computing particle flows and show the solutions for various examples as well we compute the metric distance on several anatomical configurations as measured by 0 1 v t v dt on the geodesic shortest paths

higher order mechanical systems with constraints a general mathematical theory covering higher order mechanical systems subject to constraints of arbitrary order i e depending on time positions velocities accellerations and higher derivatives is presented including higher order holonomic systems as a particular case within differential geometric setting on higher order jet bundles the concept of a mechanical system not necessarily regular or lagrangian is introduced to be a class of 2 forms equivalent with a dynamical form dynamics are then represented by means of corresponding exterior differential systems higher order constraint structure on a fibered manifold is defined to be a submanifold endowed with a distribution canonical distribution higher order chetaev bundle with help of a constraint structure a constraint force is naturally introduced higher order mechanical systems subject to different kinds of higher order constraints are then geometrically characterized and their dynamics are studied from a geometrical point of view

geometric integrators and nonholonomic mechanics a geometric derivation of nonholonomic integrators is developed it is based in the classical technique of generating functions adapted to the special features of nonholonomic systems the theoretical methodology and the integrators obtained are different from those obtained in cortes and martinez nonholonomic integrators nonlinearity 14 1365 1392 2001 in the case of mechanical systems with linear constraints a family of geometric integrators preserving the nonholonomic constraints is given

hamilton jacobi equations for nonholonomic dynamics we derive generalized hamilton jacobi equations for dynamical systems subject to linear velocity constraints as long as a solution of the generalized hamilton jacobi equation exists the action is actually minimized not just extremized

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

partial differential equations with differential constraints a geometric setting for constrained exterior differential systems on fibered manifolds with n dimensional bases is proposed constraints given as submanifolds of jet bundles locally defined by systems of first order partial differential equations are shown to carry a natural geometric structure called the canonical distribution systems of second order partial differential equations subjected to differential constraints are modeled as exterior differential systems defined on constraint submanifolds as an important particular case lagrangian systems subjected to first order differential constraints are considered different kinds of constraints are introduced and investigated lagrangian constraints constraints adapted to the fibered structure constraints arising from a co distribution semi holonomic constraints holonomic constraints

mechanical systems subjected to impulsive constraints a geometrical description of mechanical systems subjected to impulsive non holonomic constraints is given their motions are determined by means of suitable projection operators which allow us to evaluate the instantaneous jumps of momenta due to the action of impulsive reaction forces several paradigmatic examples are investigated from this viewpoint a disk falling into a plane a rolling ball hitting a wall and the collision of two balls

fast solution methods for space fractional diffusion equations we develop fast solution methods for a shifted grunwald finite difference method for steady state and time dependent space fractional diffusion equations these methods reduce the memory requirement of the finite difference scheme from o n 2 to o n and the computational complexity from o n 3 to o n log 2 n preliminary numerical example runs show the utility of these methods over the traditional direct solvers of the finite difference methods in terms of computational cost and memory requirements

hamiltonian theory of constrained impulsive motion this paper considers systems subject to nonholonomic constraints which are not uniform on the whole configuration manifold when the constraints change the system undergoes a transition in order to comply with the new imposed conditions building on previous work on the hamiltonian theory of impact we tackle the problem of mathematically describing the classes of transitions that can occur we propose a comprehensive formulation of the transition principle that encompasses the various impulsive regimes of hamiltonian systems our formulation is based on the partial symplectic formalism which provides a suitable framework for the dynamics of nonholonomic systems we pay special attention to mechanical systems and illustrate the results with several examples

on representations and integrability of mathematical structures in energy conserving physical systems in the present paper we elaborate on the underlying hamiltonian structure of interconnected energy conserving physical systems it is shown that a power conserving interconnection of port controlled generalized hamiltonian systems leads to an implicit generalized hamiltonian system and a power conserving partial interconnection to an implicit port controlled hamiltonian system the crucial concept is the notion of a generalized dirac structure defined on the space of energy variables or on the product of the space of energy variables and the space of flow variables in the port controlled case three natural representations of generalized dirac structures are treated necessary and sufficient conditions for closedness or integrability of dirac structures in all three representations are obtained the theory is applied to implicit port controlled generalized hamiltonian systems and it is shown that the closedness condition for the dirac structure leads to strong conditions on the input vector fields

on rainbow domination numbers of graphs given a graph g and a set of t colors assume that we assign an arbitrary subset of these colors to each vertex of g if we require that each vertex to which an empty set is assigned has in its neighborhood all t colors then this assignment is called a t rainbow dominating function of the graph g the corresponding invariant c r t g which is the minimum sum of numbers of assigned colors over all vertices of g is called the t rainbow domination number of g in this paper bounds for the t rainbow domination number of an arbitrary graph for an arbitrary positive integer t are given the 3 rainbow domination numbers of several classes of graphs such as paths cycles and the generalized petersen graphs p n k are investigated the 3 rainbow domination number of p n 1 is determined and the upper bounds for p n 2 and p n 3 are provided by computer search we obtain that the upper bounds for p n 2 match their exact values for n 17

variational calculus with constraints on general algebroids variational calculus on a vector bundle e equipped with a structure of a general algebroid is developed together with the corresponding analogs of euler lagrange equations constrained systems are introduced in the variational and geometrical settings the constrained euler lagrange equations are derived for analogs of holonomic vakonomic and nonholonomic constraints this general model covers the majority of first order lagrangian systems which are present in the literature and reduces to the standard variational calculus and the euler lagrange equations in classical mechanics for e tm

control by interconnection and standard passivity based control of port hamiltonian systems the dynamics of many physical processes can be suitably described by port hamiltonian ph models where the importance of the energy function the interconnection pattern and the dissipation of the system is underscored to regulate the behavior of ph systems it is natural to adopt a passivity based control pbc perspective where the control objectives are achieved shaping the energy function and adding dissipation in this paper we consider the pbc techniques of control by interconnection cbi and standard pbc in cbi the controller is another ph system connected to the plant through a power preserving interconnection to add up their energy functions while in standard pbc energy shaping is achieved via static state feedback in spite of the conceptual appeal of formulating the control problem as the interaction of dynamical systems the current version of cbi imposes a severe restriction on the plant dissipation structure that stymies its practical application on the other hand standard pbc which is usually derived from a uninspiring and non intuitive ldquopassive output generationrdquo viewpoint is one of the most successful controller design techniques the main objectives of this paper are 1 to extend the cbi method to make it more widely applicable in particular to overcome the aforementioned dissipation obstacle 2 to show that various popular variants of standard pbc can be derived proceeding from a unified perspective 3 to establish the connections between cbi and standard pbc proving that the latter is obtained restricting the former to a suitable subset providing a nice geometric interpretation to standard pbc and comparing the size of the set of ph plants for which they are applicable

hamilton jacobi theory for degenerate lagrangian systems with holonomic and nonholonomic constraints we extend hamilton jacobi theory to lagrange dirac or implicit lagrangian systems a generalized formulation of lagrangian mechanics that can incorporate degenerate lagrangians as well as holonomic and nonholonomic constraints we refer to the generalized hamilton jacobi equation as the dirac hamilton jacobi equation for non degenerate lagrangian systems with nonholonomic constraints the theory specializes to the recently developed nonholonomic hamilton jacobi theory we are particularly interested in applications to a certain class of degenerate nonholonomic lagrangian systems with symmetries which we refer to as weakly degenerate chaplygin systems that arise as simplified models of nonholonomic mechanical systems these systems are shown to reduce to non degenerate almost hamiltonian systems i e generalized hamiltonian systems defined with non closed two forms accordingly the dirac hamilton jacobi equation reduces to a variant of the nonholonomic hamilton jacobi equation associated with the reduced system we illustrate through a few examples how the dirac hamilton jacobi equation can be used to exactly integrate the equations of motion

the mechanics and control for multi particle systems this paper deals with the mechanics and control for multi particle systems from a geometric point of view the centre of mass system is viewed as a principal fibre bundle with structure group so 3 the base space of which is called the internal or shape space a natural connection and a natural riemannian metric are both defined on the centre of mass system the equations of motion for the multi particle system are derived in the lagrangian formalism adapted to the bundle structure and then reduced with the conserved total angular momentum in contrast with this the control problem is studied with non holonomic constraints i e with the vanishing total angular momentum and equations of motion are determined for an optimally controlled multi particle system the resultant equations derived in each of the mechanical and control systems are to be compared

efficient algorithms for the minimum connected domination on trapezoid graphs given the trapezoid diagram the problem of finding the minimum cardinality connected dominating set in trapezoid graphs was solved in o m n time y d liang steiner set and connected domination in trapezoid graphs inform process lett 56 2 1995 101 108 kohler e kohler connected domination and dominating clique in trapezoid graphs discr appl math 99 2000 91 110 recently improved this result to o n time for the vertex weighted case the problem of finding the minimum weighted connected dominating set in trapezoid graphs can be solved in o m nlogn time anand srinivasan m s chang k madhukar c pandu rangan efficient algorithms for the weighted domination problems on trapezoid graphs manuscript 1996 herein n m denotes the number of vertices edges of the trapezoid graph in this paper we show a different approach for the problem of finding the minimum cardinality connected dominating set in trapezoid graphs using o n time for finding the minimum weighted connected dominating set we show the problem can be solved in o nloglogn time

modeling and control of the timoshenko beam the distributed port hamiltonian approach the purpose of this paper is to show how the timoshenko beam can be fruitfully described within the framework of distributed port hamiltonian dph systems so that rather simple and elegant considerations can be drawn regarding both the modeling and control of this mechanical system after the dph model of the beam is introduced the control problem is discussed in particular it is shown how control approaches already presented in the literature can be unified and a new control methodology is presented and discussed this control methodology relies on the generalization to infinite dimensions of the concept of structural invariant casimir function and on the extension to distributed systems of the so called control by interconnection methodology in this way finite dimensional passive controllers can stabilize distributed parameter systems by shaping their total energy i e by assigning a new minimum in the desired equilibrium configuration that can be reached if a dissipative effect is introduced

comments on the presymplectic formalism and the theory of regular lagrangians with constraints the lagrangian formalism for systems with constraints is developed using a singular lagrangian defined in the extended tangent bundle t q r the dynamics defined by the new extended lagrangian that incorporates the constraints is studied using the formalism of the presymplectic geometry a comparative study with other geometric approaches is presented

towards a hamilton jacobi theory for nonholonomic mechanical systems in this paper we obtain a hamilton jacobi theory for nonholonomic mechanical systems the results are applied to a large class of nonholonomic mechanical systems the so called caplygin systems

the hamilton pontryagin principle and multi dirac structures for classical field theoriesa we introduce a variational principle for field theories referred to as the hamilton pontryagin principle and we show that the resulting field equations are the euler lagrange equations in implicit form second we introduce multi dirac structures as a graded analog of standard dirac structures and we show that the graph of a multisymplectic form determines a multi dirac structure we then discuss the role of multi dirac structures in field theory by showing that the implicit euler lagrange equations for fields obtained from the hamilton pontryagin principle can be described intrinsically using multi dirac structures finally we show a number of illustrative examples including time dependent mechanics nonlinear scalar fields maxwell s equations and elastostatics

fast local trust region technique for diffusion tensor registration using exact reorientation and regularization diffusion tensor imaging is widely used in brain connectivity research as more and more studies recruit large numbers of subjects it is important to design registration methods which are not only theoretically rigorous but also computationally efficient however the requirement of reorienting diffusion tensors complicates and considerably slows down registration procedures due to the correlated impacts of registration forces at adjacent voxel locations based on the diffeomorphic demons algorithm vercauteren 2009 we propose a fast local trust region algorithm for handling inseparable registration forces for quadratic energy functions the method guarantees that at any time and at any voxel location the velocity is always within its local trust region this local regularization allows efficient calculation of the transformation update with numeric integration instead of completely solving a large linear system at every iteration it is able to incorporate exact reorientation and regularization into the velocity optimization and preserve the linear complexity of the diffeomorphic demons algorithm in an experiment with 84 diffusion tensor images involving both pair wise and group wise registrations the proposed algorithm achieves better registration in comparison with other methods solving large linear systems yeo 2009 at the same time this algorithm reduces the computation time and memory demand tenfold

a generalization of chetaev s principle for a class of higher order nonholonomic constraints the constraint distribution in nonholonomic mechanics has a double role on the one hand it is a kinematic constraint that is it is a restriction on the motion itself on the other hand it is also a restriction on the allowed variations when using d alembert s principle to derive the equations of motion we will show that many systems of physical interest where d alembert s principle does not apply can be conveniently modeled within the general idea of the principle of virtual work by the introduction of both kinematic constraints and variational constraints as being independent entities this includes for example elastic rolling bodies and pneumatic tires also d alembert s principle and chetaev s principle fall into this scheme we emphasize the geometric point of view avoiding the use of local coordinates which is the appropriate setting for dealing with questions of global nature like reduction

mechanical systems with nonholonomic constraints a geometric setting for the theory of first order mechanical systems subject to general nonholonomic constraints is presented mechanical systems under consideration are not supposed to be lagrangian systems and the constraints are not supposed to be of a special form in the velocities as e g affine or linear a mechanical system is characterized by a certain equivalence class of 2 forms on the first jet prolongation of a fibered manifold the nonholonomic constraints are defined to be a submanifold of the first jet prolongation it is shown that this submanifold is canonically endowed with a distribution this distribution resp its vertical subdistribution has the meaning of generalized possible resp virtual displacements the concept of a constraint force is defined and a geometric version of the principle of virtual work is proposed from the principle of virtual work a formula for a workless constraint force is obtained a mechanical system subject nonholonomic constraints is modeled as a deformation of the original unconstrained system a direct characterization of a constrained system by means of a class of 2 forms along the canonical distribution is given and constrained equations of motion in an intrinsic form are found a geometric definition of regularity for systems under nonholonomic constraints is provided in particular the case of lagrangian systems is discussed also systems subject to holonomic constraints and nonholonomic constraints affine in the velocities are investigated within the range of the general scheme

non holonomic lagrangian and hamiltonian mechanics an intrinsic approach a geometrical approach to lagrangian and hamiltonian non holonomic dynamics is proposed the construction relies on a revisitation of the poincare cartan 1 form leading to the introduction of the concepts of lagrangian and hamiltonian pairs and to the implementation of a non holonomic legendre map the relationship with the standard extrinsic approach is outlined a unified canonical framework joining both lagrangian and hamiltonian aspects is proposed

nonholonomic control systems on riemannian manifolds this paper gives a general formulation of the theory of nonholonomic control systems on a riemannian manifold modeled by second order differential equations and using the unique riemannian connection defined by the metric the main concern is to introduce a reduction scheme replacing some of the second order equations by first order equations the authors show how constants of motion together with the nonholonomic constraints may be combined to yield such a reduction the theory is applied to a particular class of nonholonomic control systems that may be thought of as modeling a generalized rolling ball this class reduces to the classical example of a ball rolling without slipping on a horizontal plane

a differential geometric setting for mixed first and second order ordinary differential equations a geometrical framework is presented for modelling general systems of mixed first and second order ordinary differential equations in contrast to our earlier work on non holonomic systems the first order equations are not regarded here as a priori given constraints two nonlinear parametrized connections appear in the present framework in a symmetrical way and they induce a third connection via a suitable fibred product the space where solution curves of the given differential equations live singles out a specific projection among the many fibrations in the general picture a large part of the paper is about the development of intrinsic tools tensor fields and derivations for an adapted calculus along a major issue concerns the extent to which the usual construction of a linear connection associated with second order equations fails to work in the presence of coupled first order equations an application of the ensuing calculus is presented

a constraint algorithm for singular lagrangians subjected to nonholonomic constraints we construct a constraint algorithm for singular lagrangian systems subjected to nonholonomic constraints which generalizes that of dirac for constrained hamiltonian systems

a geometric approach to the gibbs appell equations in lagrangian mechanics we present a geometrical derivation of the gibbs appell equations in constrained lagrangian mechanics using the affine structures existing in the spaces of velocities and accelerations the theoretical basis is the gauss principle of least constraint and the necessary conditions for the minimum expressed in terms of the gibbs function give rise to the gibbs appell equations we give a second order formulation of the constraint equation and using the concept of moore penrose inverse we provide an explicit intrinsic construction of the projector on the space of the effective or non ideal forces the gibbs appell equations are expressed in an intrinsic way in terms of the fibre derivative of the gibbs function and the effective part of the applied force

reduction of some classical non holonomic systems with symmetry two types of nonholonomic systems with symmetry are treated i the configuration space is a total space of a g principal bundle and the constraints are given by a connection ii the configuration space is g itself and the constraints are given by left invariant forms the proofs are based on the method of quasicoordinates in passing a derivation of the maurer cartan equations for lie groups is obtained simple examples are given to illustrate the algorithmical character of the main results

jet methods in nonholonomic mechanics classical nonholonomic mechanical systems are studied whose evolution space is a fiber bundle m r the framework is that of jet spaces in which the geometrical meaning of the theory emerges clearly for systems with linear constraints a new interpretation is given of appell s equations as first order differential equations associated with a suitable vector field the d alembert principle is formulated in an appropriate way to be generalized to systems with nonlinear constraints the equivalence between the equations of motion arising from this generalization the ones set up on gauss principle of least constraint and hertz s principle of least curvature is established

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

geometric formulation of mechanical systems subjected to time dependent one sided constraints a geometrical setting for time dependent impulsive constraints is given this description permits us to treat in a systematic way a wide family of mechanical systems subjected to time dependent constraints like moving walls etc if the impulsive constraints remain then an affine projector can be defined giving the new initial data in terms of the old ones several examples are discussed

generalized nonholonomic mechanics servomechanisms and related brackets it is well known that nonholonomic systems obeying d alembert s principle are described on the hamiltonian side after using the legendre transformation by the so called almost poisson brackets in this paper we define the lagrangian and hamiltonian sides of a class of generalized nonholonomic systems gnhs obeying a generalized version of d alembert s principle such as rubber wheels like some simplified models of pneumatic tires and certain servomechanisms like the controlled inverted pendulum and show that corresponding equations of motion can also be described in terms of a bracket we present essentially all possible brackets in terms of which the mentioned equations can be written down which include the brackets that appear in the literature and point out those if any that are naturally related to each system in particular we show there always exists a leibniz bracket related to a gnhs and conversely that every leibniz system is a gnhs the control of the inverted pendulum on a cart

the geometrical theory of constraints applied to the dynamics of vakonomic mechanical systems the vakonomic bracket a vakonomic mechanical system can be alternatively described by an extended lagrangian using the lagrange multipliers as new variables since this extended lagrangian is singular the constraint algorithm can be applied and a dirac bracket giving the evolution of the observables can be constructed

non holonomic lagrangian systems in jet manifolds a geometrical setting in terms of jet manifolds is developed for time dependent non holonomic lagrangian systems an almost product structure on the evolution space is constructed in such a way that the constrained dynamics is obtained by projection of the free dynamics a constrained poincare cartan 2 form is defined if the non holonomic system is singular a constraint algorithm is constructed special attention is devoted to caplygin systems and a reduction theorem is proved

constraints in hamiltonian time dependent mechanics the key point of the study of constraints in hamiltonian time dependent mechanics lies in the fact that a poisson structure does not provide dynamic equations and a poisson bracket of constraints with a hamiltonian is ill defined we describe hamiltonian dynamics in terms of hamiltonian forms and connections on the vertical cotangent bundle v q r seen as a momentum phase space a poisson bracket v on v q is induced by the canonical poisson bracket t on the cotangent bundle t q with v an algebra of first and second class time dependent constraints is described but we use the pull back of the evolution equation onto t x and the bracket t in order to extend the constraint algorithm to time dependent constraints the case of lagrangian constraints of a degenerate almost regular lagrangian is studied in detail one can assign to this lagrangian l a set of hamiltonian forms which are not necessarily degenerate such that any solution of the corresponding hamilton equations which lives in the lag

reduction of constrained systems with symmetries a general model is proposed for constrained dynamical systems on a symplectic manifold which covers among others the description of lagrangian and hamiltonian systems with nonholonomic constraints and the canonical description of mechanical systems with a singular lagrangian the reduction properties of these systems in the presence of symmetry are investigated within this general framework

iper ideal kinetic constraints in continuum mechanics a geometrical approach to lagrangian formulation of continuum mechanics in the language of jet bundles theory is proposed the resulting environment allows one to deal with a particular class of kinetic constraints through a suitable implementation of gauss principle a variational formulation of systems subject to such constraints is also outlined two explicit examples are given

nonholonomic constraints in time dependent mechanics the constraint reaction force of ideal nonholonomic constraints in time dependent mechanics on a configuration bundle q r is obtained using the vertical extension of hamiltonian formalism to the vertical tangent bundle vq of q r the hamiltonian of a nonholonomic constrained system is constructed the present setting is more general than the one usually considered in the literature on nonholonomic mechanics