lagrangian hamiltonian unified formalism for field theory the rusk skinner formalism was developed in order to give a geometrical unified formalism for describing mechanical systems it incorporates all the characteristics of lagrangian and hamiltonian descriptions of these systems including dynamical equations and solutions constraints legendre map evolution operators equivalence etc in this work we extend this unified framework to first order classical field theories and show how this description comprises the main features of the lagrangian and hamiltonian formalisms both for the regular and singular cases this formulation is a first step toward further applications in optimal control theory for partial differential equations

differential geometry palatini gravity and reduction the present article deals with a formulation of the so called vacuum palatini gravity as a general variational principle in order to accomplish this goal some geometrical tools related to the geometry of the bundle of connections of the frame bundle lm are used a generalization of lagrange poincare reduction scheme to these types of variational problems allows us to relate it with the einstein hilbert variational problem relations with some other variational problems for gravity found in the literature are discussed

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

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

discrete vakonomic mechanics discretizations of vakonomic mechanics are developed by using the methodology of variational integrators the resulting algorithms are symplectic in some sense that we will explain and preserve the momentum map associated with a lie group of symmetries a specific example illustrating the theory is tested numerically

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

optimal control of underactuated mechanical systems a geometric approach in this paper we consider a geometric formalism for optimal control of underactuated mechanical systems our techniques are an adaptation of the classical skinner and rusk approach for the case of lagrangian dynamics with higher order constraints we study a regular case where it is possible to establish a symplectic framework and as a consequence to obtain a unique vector field determining the dynamics of the optimal control problem these developments will allow us to develop a new class of geometric integrators based on discrete variational calculus

heisenberg langevin and current equations via the recurrence relations approach some years ago the heisenberg equation of motion was formally solved by the recurrence relations approach it is shown here that the langevin equation represents a structural property of the recurrence relations the langevin equation is useful for studying the time evolution of the current the resulting current current correlation function is compared with luttinger s phenomenological theory geometric interpretations are made for the conductivity and the dielectric function

co isotropic and legendre lagrangian submanifolds and conformal jacobi morphisms the notion of a co isotropic and legendre lagrangian submanifold of a jacobi manifold is given a characterization of conformal jacobi morphisms and conformal jacobi infinitesimal transformations is obtained as co isotropic and legendre lagrangian submanifolds of jacobi manifolds

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

on the legendre map in higher order field theories the authors show how the construction of a cartan form in higher order field theories defines a legendre map and how the regularity of this map may be described in terms of a sequence of maps between intermediate phase spaces they also show how semi holonomic jets are important in this context they allow the definition of regularity to be applied without ambiguity to the lagrangian itself and they permit the specification of a unique cartan form legendre map and covariant phase space for second order lagrangians

new relations between hamiltonian and lagrangian constraints different procedures for obtaining all the lagrangian constraints from the hamiltonian constraints are studied the connection of arbitrary functions in both hamiltonian and lagrangian formalisms is also studied as a consequence the authors give a way to pass directly from the final hamilton dirac equations to the euler lagrange ones

lagrangian submanifolds and higher order mechanical systems higher order lagrangian and hamiltonian systems time dependent or independent are interpreted in terms of lagrangian submanifolds of sympletic higher order tangent bundles the relation between both formalisms is given

irreversible conversion of graphs given a graph g a function f v g z and an initial 0 1 vertex labelling c 1 v g 0 1 we study an iterative 0 1 vertex labelling process on g where in each round every vertex v never changes its label from 1 to 0 and changes its label from 0 to 1 if at least f v neighbours have label 1 such processes model opinion disease spreading or fault propagation and have been studied under names such as irreversible threshold majority processes in a large variety of contexts our contributions concern computational aspects related to the minimum cardinality irr f g of sets of vertices with initial label 1 such that during the process on g all vertices eventually change their label to 1 such sets are known as irreversible conversion sets dynamic irreversible monopolies or catastrophic fault patterns answering a question posed by dreyer and roberts p a dreyer jr f s roberts irreversible k threshold processes graph theoretical threshold models of the spread of disease and of opinion discrete appl math 157 2009 1615 1627 we prove a hardness result for irr f g where f v 2 for every email protected v g furthermore we describe a general reduction principle for irr f g which leads to efficient algorithms for graphs with simply structured blocks such as trees and chordal graphs

overdetermined elliptic problems in topological disks abstract we introduce a method based on the poincare hopf index theorem to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk applications to some well known nonlinear elliptic pdes are provided our result can be seen as the analogue of hopf s uniqueness theorem for constant mean curvature spheres but for the general analytic context of overdetermined elliptic problems

simple mechanical control systems with constraints we apply some recently developed control theoretic techniques to the analysis of a class of mechanical systems with constraints certain simple aspects of the theory of affine connections play an important part in our presentation the necessary background is presented in order to illustrate how the methods may be applied the bulk of this paper is devoted to a detailed analysis of some examples of nonholonomic mechanical control systems we look at the heisenberg system the upright rolling disk the roller racer and the snakeboard

nonholonomic motion planning steering using sinusoids methods for steering systems with nonholonomic c onstraints between arbitrary configurations are investigated suboptimal trajectories are derived for systems that are not in canonical form systems in which it takes more than one level of bracketing to achieve controllability are considered the trajectories use sinusoids at integrally related frequencies to achieve motion at a given bracketing level a class of systems that can be steered using sinusoids claimed systems is defined conditions under which a class of two input systems can be converted into this form are given

d alembert lagrange analytical dynamics for nonholonomic systems the d alembert lagrange principle dlp is designed primarily for dynamical systems under ideal geometric constraints although it can also cover linear velocity constraints its application to nonlinear kinematic constraints has so far remained elusive mainly because there is no clear method whereby the set of linear conditions that restrict the virtual displacements can be easily extracted from the equations of constraint on recognition that the commutation rule traditionally accepted for velocity displacements in lagrangian dynamics implies displaced states that do not satisfy the kinematic constraints we show how the property of possible displaced states can be utilized ab initio so as to provide an appropriate set of linear auxiliary conditions on the displacements which can be adjoined via lagrange s multipliers to the d alembert lagrange equation to yield the equations of state and also new transpositional relations for nonholonomic systems the equations of state so obtained for systems under general nonlinear velocity and acceleration constraints are shown to be identical with those derived in appendix a from the quite different gauss principle the present advance therefore solves a long outstanding problem on the application of dlp to ideal nonholonomic systems and as an aside provides validity to axioms as the chetaev rule previously left theoretically unjustified a more general transpositional form of the boltzmann hamel equation is also obtained

unified formalism for nonautonomous mechanical systems we present a unified geometric framework for describing both the lagrangian and hamiltonian formalisms of regular and non regular time dependent mechanical systems which is based on the approach of skinner and rusk 18 the dynamical equations of motion and their compatibility and consistency are carefully studied making clear that all the characteristics of the lagrangian and the hamiltonian formalisms are recovered in this formulation as an example it is studied a semidiscretization of the nonlinear wave equation proving the applicability of the proposed formalism

a hopf differential for constant mean curvature surfaces in s2 r and h2 r a basic tool in the theory of constant mean curvature cmc surfaces in space forms is the holomorphic quadratic differential discovered by h hopf in this paper we generalize this differential to immersed cmc surfaces in the product spaces s r and h r and prove a corresponding result about the geometry of immersed cmc spheres s in these target spaces

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

canonical structures for dispersive waves in shallow water the canonical hamiltonian structure of the equations of fluid dynamics obtained in the boussinesq approximation are considered new variational formulations of these equations are proposed and it is found that as in the case of the kdv equation and the equations governing long waves in shallow water they are degenerate lagrangian systems therefore in order to cast these equations into canonical form it is again necessary to use dirac s theory of constraints it is found that there are primary and secondary constraints which are second class and it is possible to construct the hamiltonian in terms of canonical variables among the examples of boussinesq equations that are discussed are the equations of whitham broer kaup which kupershmidt has recently expressed in symmetric form and shown to admit tri hamiltonian structure

reductions of topologically massive gravity ii first order realizations of second order lagrangians second order degenerate clement and sarioglu tekin lagrangians are casted into forms of various first order lagrangians the structures of the iterated tangent bundle and acceleration bundle are presented as a suitable geometric framework hamiltonian analyses of these equivalent formalisms are performed by means of the dirac bergmann constraint algorithm all formulations are shown to possess only second class constraints

convex and isometric domination of weak dominating pair graphs abstract a set d of vertices in a graph g is a dominating set if every vertex of g which is not in d has a neighbor in d a set of vertices d in g is convex respectively isometric if all vertices in all shortest paths respectively all vertices in one of the shortest paths between any two vertices in d lie in d the problem of finding a minimum convex dominating respectively isometric dominating set is considered in this paper from algorithmic point of view for the class of weak dominating pair graphs i e the graphs that contain a dominating pair which is a pair of vertices x y v g such that vertices of any path between x and y form a dominating set we present an efficient algorithm that finds a minimum isometric dominating set of such a graph on the other hand we prove that even if one restricts to weak dominating pair graphs that are also chordal graphs the problem of deciding whether there exists a convex dominating set bounded by a given arbitrary positive integer is np complete by further restricting the class of graphs to chordal dominating pair graphs i e the chordal graphs in which every connected induced subgraph has a dominating pair we are able to find a polynomial time algorithm that determines the minimum size of a convex dominating set of such a graph

constrained variational calculus for higher order classical field theories we develop an intrinsic geometrical setting for higher order constrained field theories as a main tool we use an appropriate generalization of the classical skinner rusk formalism some examples of applications are studied in particular to the geometrical description of optimal control theory for partial differential equations

hamiltonian dynamics and constrained variational calculus continuous and discrete settings the aim of this paper is to study the relationship between hamiltonian dynamics and constrained variational calculus we describe both using the notion of lagrangian submanifolds of convenient symplectic manifolds and using the so called tulczyjew triples the results are also extended to the case of discrete dynamics and nonholonomic mechanics interesting applications to the geometrical integration of hamiltonian systems are obtained

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

abnormal minimizers this paper constructs the first example of a singular abnormal minimizer for the lagrange problem with linear velocity constraints and quadratic definite lagrangian or equivalently for an optimal control system of linear controls with k controls n states and a running cost function that is quadratic positive definite in the controls in the example k 2 n 3 and the system is completely controllable the example is stable if both the control law and cost are perturbed the singular minimizer persists its importance is due in part to the fact that it is a counterexample to a theorem that has appeared several times in the differential geometry literature there the problem is called the problem of finding minimizing sub riemannian geodesics and it has been claimed that all minimizers are normal pontryagin extremals the mathematical theory of optimal processes wiley interscience new york 1962 if the number of states equals the number of controls then the problem is that of finding riemannian geodesics the main difficulty is proving minimality to do this the length cost of the abnormal is compared with all competing normal extremals a detailed asymptotic analysis of the differential equations governing the normals shows that they are all longer

equivalence of the dynamics of nonholonomic and variational nonholonomic systems for certain initial data in this paper we discuss the necessary and sufficient conditions for the equivalence of the dynamics of nonholonomic mechanics and variational nonholonomic vakonomic dynamics for certain initial conditions we derive a priori results for identifying equivalence and specializing to abelian chaplygin systems prove that equivalence results if and only if the constrained nonholonomic equations are lagrangian we eliminate the need to solve the variational nonholonomic problem when checking equivalence by obtaining explicit formulae for the system s multipliers and then derive conditions under which the multiplier free lagrangian gives equivalence of the dynamics we consider nonholonomic systems possessing invariant measures showing when equivalence and hamiltonization are the same we also derive conditions under which measure preserving systems exhibit equivalence we apply the results to many of the known nonholonomic systems

a geometric setting for higher order dirac bergmann theory of constraints hamilton equations for degenerate time dependent and generally higher order mechanical systems are considered and their geometric meaning is analyzed the local dynamics of constrained systems is clarified and an algorithm for finding it is presented

integrability of implicit differential equations the problem of integrability of differential equations is discussed examples and integrability criteria are given an algorithm for extracting the integrable part of an implicit differential equation is formulated a procedure for generating a class of submanifolds of the cotangent bundle is defined this procedure is then used for generating implicit differential equations in the phase space of a mechanical system integrability criteria of such equations are established and an extraction algorithm is formulated

on the quantization of constrained generalized dynamics a special class of degenerate second order lagrangians those that differ from a nondegenerate first order lagrangian by a total time derivative or a four divergence of a function of both the coordinates and velocities is studied in detail using dirac s theory of constrained systems it is shown that the canonical quantization starting from the second order lagrangian leads to the same physical results as those obtained from the nondegenerate first order lagrangian then some incorrect results and misleading arguments encountered in the literature on the subject are clarified

geometric lagrangian approach to first order systems and applications the geometric theory of presymplectic systems is developed to study both the lagrangian and hamiltonian formulations of a system described by a lagrangian linear in the velocities the results are used to study some related problems for second order differential equations and regular lagrangians

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

on the caratheodory number of interval and graph convexities inspired by a result of caratheodory uber den variabilitatsbereich der fourierschen konstanten von positiven harmonischen funktionen rend circ mat palermo 32 1911 193 217 the caratheodory number of a convexity space is defined as the smallest integer k such that for every subset u of the ground set v and every element u in the convex hull of u there is a subset f of u with at most k elements such that u in the convex hull of f we study the caratheodory number for generalized interval convexities and for convexity spaces derived from finite graphs we establish structural properties bounds and hardness results

sub finslerian metric associated to an optimal control system the problem of minimizing the cost functional of an optimal control system through the use of constrained variational calculus is a generalization of the geodetic problem in riemannian geometry in the framework of a geometric formulation of optimal control we define a metric structure associated to the optimal control system on the enlarged space of state and time variables such that the minimal length curves of the metric are the optimal solutions of the system a twofold generalization of metric structure is applied considering finslerian type metrics as well as allowed and forbidden directions like in sub riemannian geometry free null hamiltonian or fixed final parameter problems are identified with constant energy leaves and the restriction of the metric to these leaves gives way to a family of metric structures on the usual state manifold

regularity properties of fiber derivatives associated with higher order mechanical systems the aim of this work is to study fiber derivatives associated to lagrangian and hamiltonian functions describing the dynamics of a higher order autonomous dynamical system more precisely given a function in t t k 1 q we find necessary and sufficient conditions for such a function to describe the dynamics of a kth order autonomous dynamical system thus being a kth order hamiltonian function then we give a suitable definition of hyper regularity for these higher order hamiltonian functions in terms of their fiber derivative in addition we also study an alternative characterization of the dynamics in lagrangian submanifolds in terms of the solutions of the higher order euler lagrange equations

regularity aspects and hamiltonization of non holonomic systems we discuss various aspects of the transition from lagrangian to hamiltonian equations for systems with general nonlinear non holonomic constraints the emphasis is first on constructing the reduced dynamics on the constraint submanifold and then trying to start a hamiltonization procedure from there we prove a theorem concerning the regularity which is required to obtain a unique second order dynamics on the constraint submanifold and we show that the same condition allows the transition to a hamiltonian picture throughout the analysis different degrees of generality are discussed

rotational hypersurfaces of prescribed mean curvature abstract we use a phase space analysis to give some classification results for rotational hypersurfaces in r n 1 whose mean curvature is given as a prescribed function of its gauss map for the case where the prescribed function is an even function in s n we show that a delaunay type classification holds for this class of hypersurfaces we also exhibit examples showing that the behavior of rotational hypersurfaces of prescribed non constant mean curvature is much richer than in the constant mean curvature case

morse families in optimal control problems we geometrically describe optimal control problems in terms of morse families in the hamiltonian framework these geometric structures allow us to recover the classical first order necessary conditions for optimality and the starting point to run an integrability algorithm moreover the integrability algorithm is adapted to optimal control problems in such a way that the trajectories originated by discontinuous controls are also obtained from the hamiltonian viewpoint we obtain the equations of motion for optimal control problems in the lagrangian formalism by means of a proper lagrangian submanifold singular optimal control problems and overdetermined ones are also studied in the paper

symplectic reduction of higher order lagrangian systems with symmetry one can develop a symplectic reduction procedure for higher order lagrangian systems with symmetry the reconstruction procedure of the dynamics is also studied and an application spinning particle is given at the end of the work

a dynamical formalism of singular lagrangian system with higher derivatives the singular lagrangian system with higher derivatives is analyzed with the aid of the ostrogradski transformation and the dirac formalism the formulation of canonical theory is developed so that the equivalence between the lagrange formalism and the hamilton one is maintained as a practical example the acceleration dependent potentials appearing in the lagrangian of two point particles interacting gravitationally are dealt with and the equivalence between the two hamiltonians that follow from the two lagrangians which are related by the coordinate transformations is shown it is also shown when the constraints are all first class that a consistent generator of gauge transformation is constructed typical examples are given

newton s law and integrability of nonholonomic systems in this paper we consider nonholonomic control systems on riemannian manifolds such systems evolve on subbundles of tangent bundles defined by the nonholonomic constraints this paper promotes the view of such systems as the restriction to the nonholonomic subbundle of newton law type problems on the entire tangent bundle defined by in general non riemannian connections these connections should be related to specific geometric properties of the nonholonomic system we introduce a particular class of connections and demonstrate the richness of the class through four examples the rolling ball the constrained particle the rolling penny and the generalized rolling ball this class of connections is strongly related to questions of integrability of the original nonholonomic system this in turn provides additional insight into the relation between nonholonomic control systems formulated as kinematic equations and those that are formulated as the full dynamic equations

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

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 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

a generalized lagrange equation in implicit form for non conservative mechanics different geometric formulations are obtained for a generalized lagrange equation non reducible to normal form and encompassing non conservative dynamics

equivalence between the lagrangian and hamiltonian formalism for constrained systems the equivalence between the lagrangian and hamiltonian formalism is studied for constraint systems a procedure to construct the lagrangian constraints from the hamiltonian constraints is given those hamiltonian constraints that are first class with respect to the hamiltonian constraints produce lagrangian constraints that are fl projectable

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

on the parameterized complexity of the geodesic hull number abstract several recent papers obtained complexity results regarding the geodesic hull number hn gd g of a graph g in this paper we prove that determining whether hn gd g k is w 2 hard parameterized by k in diameter two graphs and is w 1 hard parameterized by tw k where tw is the treewidth of g despite this for graphs with bounded treewidth tw we prove that hn gd g is computable in polynomial time o tw 2 tw 5 n 2 tw 5

symmetries and infinitesimal symmetries of singular differential equations the aim of this paper is to study symmetries of linearly singular differential equations namely equations that cannot be written in normal form because the derivatives are multiplied by a singular linear operator the concept of geometric symmetry of a linearly singular differential equation is introduced as a transformation that preserves the geometric data that define the problem it is proved that such symmetries are essentially equivalent to dynamic symmetries that is transformations mapping solutions into solutions similar results are given for infinitesimal symmetries to study the invariance of several objects under the flows of vector fields a careful study of infinitesimal variations is performed with a special emphasis on infinitesimal vector bundle automorphisms

singular lagrangians with higher derivatives the hamiltonian formalism for systems with singular lagrangians of the second order is constructed a new method is proposed for obtaining the equations of motion in the phase space for theories with singular lagrangians and the connection of the lagrangian and hamiltonian descriptions is traced in detail as an example a generalisation of the relativistic point action is considered it involves both the length and curvature of the point trajectory in spacetime

reduction of constrained mechanical systems and stability of relative equilibria a mechanical system with perfect constraints can be described under some mild assumptions as a constrained hamiltonian system m h d w m thephase space is a symplectic manifold h thehamiltonian a smooth function onm d theconstraint submanifold a submanifold ofm andw theprojection bundle a vector sub bundle oft d m the reduced tangent bundle alongd we prove that when these data satisfy some suitable conditions the time evolution of the system is governed by a well defined differential equation ond we define constrained hamiltonian systems with symmetry and prove a reduction theorem application of that theorem is illustrated on the example of a convex heavy body rolling without slipping on a horizontal plane two other simple examples show that constrained mechanical systems with symmetry may have an attractive or repulsive set of relative equilibria

higher order conditions for singular lagrangian dynamics for a kth order singular lagrangian the presymplectic equation plus the different mth order differential equation conditions that can be considered in the lagrangian space yield different dynamics which are studied this leads to a new classification of the constraints this study is also performed in the intermediate formalisms which can be defined between the lagrangian and the hamiltonian ones the corresponding classification schemes are then related

isoperimetric domains of large volume in homogeneous three manifolds abstract given a non compact simply connected homogeneous three manifold x and a sequence n n of isoperimetric domains in x with volumes tending to infinity we prove that as n 1 the radii of the n tend to infinity 2 the ratios area n vol n converge to the cheeger constant ch x which we also prove to be equal to 2 h x where h x is the critical mean curvature of x 3 the values of the constant mean curvatures h n of the boundary surfaces n converge to 1 2 ch x furthermore when ch x is positive we prove that for n large n is well approximated in a natural sense by the leaves of a certain foliation of x where every leaf of the foliation is a surface of constant mean curvature h x

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

higher order lagrangian systems geometric structures dynamics and constraints in order to study the connections between lagrangian and hamiltonian formalisms constructed from a perhaps singular higher order lagrangian some geometric structures are constructed intermediate spaces between those of lagrangian and hamiltonian formalisms partial ostrogradski s transformations and unambiguous evolution operators connecting these spaces are intrinsically defined and some of their properties studied equations of motion constraints and arbitrary functions of lagrangian and hamiltonian formalisms are thoroughly studied in particular all the lagrangian constraints are obtained from the hamiltonian ones once the gauge transformations are taken into account the true number of degrees of freedom is obtained both in the lagrangian and hamiltonian formalisms and also in all the intermediate formalisms herein defined

a hopf theorem for non constant mean curvature and a conjecture of a d alexandrov we prove a uniqueness theorem for immersed spheres of prescribed non constant mean curvature in homogeneous three manifolds in particular this uniqueness theorem proves a conjecture by a d alexandrov about immersed spheres of prescribed weingarten curvature in mathbb r 3 for the special but important case of prescribed mean curvature as a consequence we extend the classical hopf uniqueness theorem for constant mean curvature spheres to the case of immersed spheres of prescribed antipodally symmetric mean curvature in mathbb r 3

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

unambiguous formalism for higher order lagrangian field theories the aim of this paper is to propose an unambiguous intrinsic formalism for higher order field theories which avoids the arbitrariness in the generalization of the conventional description of field theories and implies the existence of different cartan forms and legendre transformations we propose a differential geometric setting for the dynamics of a higher order field theory based on the skinner and rusk formalism for mechanics this approach incorporates aspects of both the lagrangian and the hamiltonian description since the field equations are formulated using the lagrangian on a higher order jet bundle and the canonical multisymplectic form on its affine dual as both of these objects are uniquely defined the skinner rusk approach has the advantage that it does not suffer from the arbitrariness in conventional descriptions the result is that we obtain a unique and global intrinsic version of the euler lagrange equations for higher order field theories several examples illustrate our construction

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

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

skinner rusk unified formalism for optimal control systems and applications a geometric approach to time dependent optimal control problems is proposed this formulation is based on the skinner and rusk formalism for lagrangian and hamiltonian systems the corresponding unified formalism developed for optimal control systems allows us to formulate geometrically the necessary conditions given by a weak form of pontryagin s maximum principle provided that the differentiability with respect to controls is assumed and the space of controls is open furthermore our method is also valid for implicit optimal control systems and in particular for the so called descriptor systems optimal control problems including both differential and algebraic equations

controllability and stabilizability of a class of systems with higher order nonholonomic constraints a theoretical framework is established for the control of higher order nonholonomic systems defined as systems that satisfy higher order nonintegrable constraints a model for such systems is developed in terms of differential algebraic equations defined on a higher order tangent bundle a number of control theoretic properties such as nonintegrability controllability and stabilizability are presented higher order nonholonomic systems are shown to be strongly accessible and under certain conditions small time locally controllable at any equilibrium the applicability of the theoretical development is illustrated through a third order nonholonomic system example a planar ppr robot manipulator subject to a jerk constraint in particular it is shown that although the system is not asymptotically stabilizable to a given equilibrium configuration using a time invariant continuous feedback it is strongly accessible and small time locally controllable at any equilibrium

constrained hamiltonian systems as implicit differential equations generalized hamiltonian systems derived from degenerate lagrangians are presented as implicit differential equations rather than families of hamiltonian vector fields restricted to constraint subsets of the phase space systems considered are more general than the ones described by dirac they are generated by morse families of functions and not by constrained hamiltonians integrability of such systems is analysed examples are given

reductions of topologically massive gravity i hamiltonian analysis of second order degenerate lagrangians we present skinner rusk and hamiltonian formalisms of second order degenerate clement and sarioglu tekin lagrangians the dirac bergmann constraint algorithm is employed to obtain hamiltonian realizations of lagrangian theories the gotay nester hinds algorithm is used to investigate skinner rusk formalisms of these systems

inapproximability results for graph convexity parameters in this paper we prove several inapproximability results on the p 3 convexity and the geodesic convexity in graphs we prove that determining the p 3 hull number and the geodetic hull number are apx hard problems we prove that the caratheodory number the radon number and the convexity number of both convexities are o n 1 e inapproximable in polynomial time for every e 0 unless p np we also prove that the interval numbers of both convexities are w 2 hard and o log n inapproximable in polynomial time unless p np moreover these results hold for bipartite graphs in the p 3 convexity

singular lagrangians some geometric structures along the legendre map new geometric structures that relate the lagrangian and hamiltonian formalisms defined upon a singular lagrangian are presented several vector fields are constructed in velocity space that give new and precise answers to several topics such as the projectability of a vector field to a hamiltonian vector field the computation of the kernel of the presymplectic form of a lagrangian formalism the construction of the lagrangian dynamical vector fields and the characterization of dynamical symmetries