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

multivector fields and connections setting lagrangian equations in field theories the integrability of multivector fields in a differentiable manifold is studied then given a jet bundle j1e e m it is shown that integrable multivector fields in e are equivalent to integrable connections in the bundle e m that is integrable jet fields in j1e this result is applied to the particular case of multivector fields in the manifold j1e and connections in the bundle j1e m that is jet fields in the repeated jet bundle j1j1e in order to characterize integrable multivector fields and connections whose integral manifolds are canonical lifting of sections these results allow us to set the lagrangian evolution equations for first order classical field theories in three equivalent geometrical ways in a form similar to that in which the lagrangian dynamical equations of nonautonomous mechanical systems are usually given then using multivector fields we discuss several aspects of these evolution equations both for the regular and singular cases namely the existence and nonuniqueness o

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

efficient design with interdependent valuations we study efficient bayes nash incentive compatible mechanisms in a social choice setting that allows for informational and allocative externalities we show that such mechanisms exist only if a congruence condition relating private and social rates of information substitution is satisfied if signals are multi dimensional the congruence condition is determined by an integrability constraint and it can hold only in nongeneric cases where values are private or a certain symmetry assumption holds if signals are one dimensional the congruence condition reduces to a monotonicity constraint and it can be generically satisfied we apply the results to the study of multi object auctions and we discuss why such auctions cannot be reduced to one dimensional models without loss of generality

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 hamilton jacobi theory we develop a discrete analogue of hamilton jacobi theory in the framework of discrete hamiltonian mechanics the resulting discrete hamilton jacobi equation is discrete only in time we describe a discrete analogue of jacobi s solution and also prove a discrete version of the geometric hamilton jacobi theorem the theory applied to discrete linear hamiltonian systems yields the discrete riccati equation as a special case of the discrete hamilton jacobi equation we also apply the theory to discrete optimal control problems and recover some well known results such as the bellman equation discrete time hjb equation of dynamic programming and its relation to the costate variable in the pontryagin maximum principle this relationship between the discrete hamilton jacobi equation and bellman equation is exploited to derive a generalized form of the bellman equation that has controls at internal stages

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

multivectorial generalization of the cartan map a multivectorial generalization of the spinors to vectors cartan map is constructed with space time as a base space constructed from spinors dirac spinors and spinors pairs being particular cases onto a multivector space with two internal symmetries one of them is the fundamental space time clifford group c 1 3 and the other an isotopic space in multivectorial representation related with the more common gauge groups the algebraic properties of this map and of the operators representing observables are studied the inversion map is also presented the quaternion projection of the map is shown as are its properties and its mathematical and physical meaning the procedure presented here is compared with other spinor vector maps used in the literature

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

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

multivector field formulation of hamiltonian field theories equations and symmetries we state the intrinsic form of the hamiltonian equations of first order classical field theories in three equivalent geometrical ways using multivector fields jet fields and connections thus these equations are given in a form similar to that in which the hamiltonian equations of mechanics are usually given then using multivector fields we study several aspects of these equations such as the existence and non uniqueness of solutions and the integrability problem in particular these problems are analysed for the case of hamiltonian systems defined in a submanifold of the multimomentum bundle furthermore the existence of first integrals of these hamiltonian equations is considered and the relation between cartan noether symmetries and general symmetries of the system is discussed noether s theorem is also stated in this context both the classical version and its generalization to include higher order cartan noether symmetries finally the equivalence between the lagrangian and hamiltonian formalisms is also discussed

multisymplectic unified formalism for einstein hilbert gravity we present a covariant multisymplectic formulation for the einstein hilbert model of general relativity as it is described by a second order singular lagrangian this is a gauge field theory with constraints the use of the unified lagrangian hamiltonian formalism is particularly interesting when it is applied to these kinds of theories since it simplifies the treatment of them in particular the implementation of the constraint algorithm the retrieval of the lagrangian description and the construction of the covariant hamiltonian formalism in order to apply this algorithm to the covariant field equations they must be written in a suitable geometrical way which consists of using integrable distributions represented by multivector fields of a certain type we apply all these tools to the einstein hilbert model without and with energy matter sources we obtain and explain the geometrical and physical meaning of the lagrangian constraints and we construct the multimomentum covariant hamiltonian formalisms in both cases as a consequence of the gauge freedom and the constraint algorithm we see how this model is equivalent to a first order regular theory without gauge freedom in the case of the presence of energy matter sources we show how some relevant geometrical and physical characteristics of the theory depend on the type of source in all the cases we obtain explicitly multivector fields which are solutions to the gravitational field equations finally a brief study of symmetries and conservation laws is done in this context we present a covariant multisymplectic formulation for the einstein hilbert model of general relativity as it is described by a second order singular lagrangian this is a gauge field theory with constraints the use of the unified lagrangian hamiltonian formalism is particularly interesting when it is applied to these kinds of theories since it simplifies the treatment of them in particular the implementation of the constraint algorithm the retrieval of the lagrangian description and the construction of the covariant hamiltonian formalism in order to apply this algorithm to the covariant field equations they must be written in a suitable geometrical way which consists of using integrable distributions represented by multivector fields of a certain type we apply all these tools to the einstein hilb

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

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

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

generalized hamiltonian dynamics i formulation on t q tq the dirac bergmann generalized hamiltonian dynamics for a degenerate lagrangian system is formulated on the whitney sum t q tq of the phase space t q and the velocity space tq over the configuration space q the formulation is related to those on t q and tq some ambiguities concerning generalized dynamics that have appeared in the literature are clarified

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

the algebraic geometry of perfect and sequential equilibrium two of the most important refinements of the nash equilibrium concept for extensive form games with perfect recall are selten s 1975 perfect equilibrium and kreps and wilson s 1982 more inclusive sequential equilibrium these two equilibrium refinements are motivated in very different ways nonetheless as kreps and wilson 1982 section 7 point out the two concepts lead to similar prescriptions for equilibrium play for each particular game form every perfect equilibrium is sequential moreover for almost all assignments of payoffs to outcomes almost all sequential equilibrium strategy profiles are perfect equilibrium profiles and all sequential equilibrium outcomes are perfect equilibrium outcomes we establish a stronger result for almost all assignments of payoffs to outcomes the sets of sequential and perfect equilibrium strategy profiles are identical in other words for almost all games each strategy profile which can be supported by beliefs satisfying the rationality requirement of sequential equilibrium can actually be supported by beliefs satisfying the stronger rationality requirement of perfect equilibrium we obtain this result by exploiting the algebraic geometric structure of these equilibrium correspondences following from the fact that they are semi algebraic sets i e they are defined by finite systems of polynomial inequalities that the perfect and sequential equilibrium correspondences have this semi algebraic structure follows from a deep result from mathematical logic the tarski seidenberg theorem that this structure has important game theoretic consequences follows from deep properties of semi algebraic sets

on the extraction of the boundary conditions and the boundary ports in second order field theories in this paper we consider second order field theories in a variational setting from the variational principle the euler lagrange equations follow in an unambiguous way but it is well known that this is not true for the cartan form this also has consequences on the derivation of the boundary conditions when non trivial variations are allowed on the boundary by posing extra conditions on the set of possible boundary terms we exploit the degree of freedom in the cartan form to extract physical meaningful boundary expressions the same mathematical machinery will be applied to derive the boundary ports in a hamiltonian representation of the partial differential equations which is crucial for energy based control approaches our results will be visualized for mechanical systems such as beam and plate models in this paper we consider second order field theories in a variational setting from the variational principle the euler lagrange equations follow in an unambiguous way but it is well known that this is not true for the cartan form this also has consequences on the derivation of the boundary conditions when non trivial variations are allowed on the boundary by posing extra conditions on the set of possible boundary terms we exploit the degree of freedom in the cartan form to extract physical meaningful boundary expressions the same mathematical machinery will be applied to derive the boundary ports in a hamiltonian representation of the partial differential equations which is crucial for energy based control approaches our results will be visualized for mechanical systems such as beam and plate models

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

geometry of multisymplectic hamiltonian first order field theories in the jet bundle description of field theories multisymplectic models in particular there are several choices for the multimomentum bundle where the covariant hamiltonian formalism takes place as a consequence several proposals for this formalism can be stated and on each one of them the differentiable structures needed for setting the formalism are obtained in different ways in this work we make an accurate study of some of these hamiltonian formalisms showing their equivalence in particular the geometrical structures canonical or not needed for the hamiltonian formalism are introduced and compared and the derivation of hamiltonian field equations from the corresponding variational principle is shown in detail furthermore the hamiltonian formalism of systems described by lagrangians is performed both for the hyper regular and almost regular cases finally the role of connections in the construction of hamiltonian field theories is clarified

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

vorticity and symplecticity in multi symplectic lagrangian gas dynamics the lagrangian multi dimensional ideal compressible gas dynamic equations are written in a multi symplectic form in which the lagrangian fluid labels m i the lagrangian mass coordinates and time t are the independent variables and in which the eulerian position of the fluid element and the entropy are the dependent variables constraints in the variational principle are incorporated by means of lagrange multipliers the constraints are the entropy advection equation s t 0 the lagrangian map equation where is the fluid velocity and the mass continuity equation which has the form where is the jacobian of the lagrangian map in which and is the specific volume of the gas the internal energy per unit volume of the gas corresponds to a non barotropic gas the lagrangian is used to define multi momenta and to develop de donder weyl hamiltonian equations the de donder weyl equations are cast in a multi symplectic form the pullback conservation laws and the symplecticity conservation laws are obtained one class of symplecticity conservation laws give rise to vorticity and potential vorticity type conservation laws and another class of symplecticity laws are related to derivatives of the lagrangian energy conservation law with respect to the lagrangian mass coordinates m i we show that the vorticity symplecticity laws can be derived by a lie dragging method and also by using noether s second theorem and a fluid relabelling symmetry which is a divergence symmetry of the action we obtain the cartan poincare form describing the equations and we discuss a set of differential forms representing the equation system

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

how often do simple dynamical processes have infinitely many coexisting sinks this work concerns the nature of chaotic dynamical processes sheldon newhouse wrote on dynamical processes depending on a parameter xx 1 t xn wherex is in the plane such as might arise when studying poincare return maps for autonomous differential equations in ir3 he proved that if the system is chaotic there will very often be existing parameter values for which there are infinitely many periodic attractors coexisting in a bounded region of the plane and that such parameter values would be dense in some interval the fact that infinitely many coexisting sinks can occur brings into question the very nature of the foundations of chaotic dynamical processes we prove for an apparently typical situation that newhouse s construction yields only a set of parameter values of measure zero

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

voronoi cell patterns theoretical model and applications we use a simple fragmentation model to describe the statistical behavior of the voronoi cell patterns generated by a homogeneous and isotropic set of points in 1d and in 2d in particular we are interested in the distribution of sizes of these voronoi cells our model is completely defined by two probability distributions in 1d and again in 2d the probability to add a new point inside an existing cell and the probability that this new point is at a particular position relative to the preexisting point inside this cell in 1d the first distribution depends on a single parameter while the second distribution is defined through a fragmentation kernel in 2d both distributions depend on a single parameter the fragmentation kernel and the control parameters are closely related to the physical properties of the specific system under study we use our model to describe the voronoi cell patterns of several systems specifically we study the island nucleation with irreversible attachment the 1d car parking problem the formation of second level administrative divisions and the pattern formed by the paris metro stations

k cosymplectic manifolds and lagrangian field theories a geometrical description of classical field theories of first order is given the underlying k cosymplectic structure permits to derive the corresponding field equations

prevalence a translation invariant almost every on infinite dimensional spaces we present a measure theoretic condition for a property to hold almost everywhere on an infinite dimensional vector space with particularemphasis on function spaces such as c k and l p like the concept of lebesgue almost every on an infinite dimensional spaces our notion of prevalence is translation invariant instead of using a specific measure on the entire space we define prevalence in terms of the class of all probability measures with compact support prevalence is a more appropriate condition than the topological concepts of open and dense or generic when one desires a probabilistic result on the likelihood of a given property on a function space

on the validity of dirac s conjecture regarding first class secondary constraints two examples are presented which clarify dirac s conjecture that all first class secondary constraints generate gauge transformations in the general case the status of this conjecture is ambiguous depending crucially upon the physical interpretation of the lagrangian at hand however the author proposes a consistent general method of interpreting lagrangians such that relative to this interpretative framework dirac s conjecture holds true

mean field approximation for spacing distribution functions in classical systems we propose a mean field method to calculate approximately the spacing distribution functions p n s in one dimensional classical many particle systems we compare our method with two other commonly used methods the independent interval approximation and the extended wigner surmise in our mean field approach p n s is calculated from a set of langevin equations which are decoupled by using a mean field approximation we find that in spite of its simplicity the mean field approximation provides good results in several systems we offer many examples illustrating that the three previously mentioned methods give a reasonable description of the statistical behavior of the system the physical interpretation of each method is also discussed

lexicographic probabilities and equilibrium refinements this paper develops a decision theoretic approach to normal form refinements of nash equilibrium and provides characterizations of normal form perfect equilibrium and proper equilibrium the approach relies on a theory of single person decision making that is a non archimedean version of subjective expected utility theory copyright 1991 by the econometric society

first order equations of motion for classical mechanics a global canonical first order equation of motion is derived for any mechanical system obeying newton s second law the existence of a lagrangian is not assumed but the properties of the canonical equation are similar to those of the hamiltonian formulation the choice of map f from velocity space to phase space is not determined by the condition that the first order equation of motion be equivalent to a second order equation on configuration space and therefore is left open to be selected on the basis of other considerations the canonical equation is a covector or 1 form equation on the whitney sum t q tq and contains the second order equation condition restriction to the graph of the map f and newton s equation of motion in first order form the last is related to newton s second order equations by the consistency condition that the motions not lead off graph f in t q tq the first order equation of motion can be projected onto phase space if the map f can be inverted

numerical methods for hamiltonian pdes the paper provides an introduction and survey of conservative discretization methods for hamiltonian partial differential equations the emphasis is on variational symplectic and multi symplectic methods the derivation of methods as well as some of their fundamental geometric properties are discussed basic principles are illustrated by means of examples from wave and fluid dynamics

genericity and markovian behavior in stochastic games this paper esamines markov perfect equilibria of general finite state stochastic games our main result is that the number of such equilibria is finite for a set of stochastic game payoffs with full lebesgue measure we further discuss extensions to lower dimensional stochastie games like the alternating move game

on the generic finiteness of equilibrium outcome distributions in game forms consider nonempty finite pure strategy sets s 1 s n let s s 1 x x s n let be a finite space of outcomes let be the set of probability distributions on and let s be a function we study the conjecture that for any utility in a generic set of n tuples of utilities on there are finitely many distributions on induced by the nash equilibria of the game given by the induced utilities on s we give a counterexample refuting the conjecture for n 3 several special cases of the conjecture follow from well known theorems and we provide some generalizations of these results

spacing distribution functions for the one dimensional point island model with irreversible attachment we study the configurational structure of the point island model for epitaxial growth in one dimension in particular we calculate the island gap and capture zone distributions our model is based on an approximate description of nucleation inside the gaps nucleation is described by the joint probability density p xy n x y which represents the probability density to have nucleation at position x within a gap of size y our proposed functional form for p xy x y describes excellently the statistical behavior of the system we compare our analytical model with extensive numerical simulations our model retains the most relevant physical properties of the system

classical brst cohomology and invariant functions on constraint manifolds i given a manifoldm and a submanifoldc together with a lie groupg acting onm and leavingc invariant it is shown how the algebra ofg invariant functions onc can be described in terms of cohomology whenc is defined as the common zero level of an irreducible set of g covariant constraints the construction is independent of any additional structures such as e g a symplectic structure onm and therefore it provides a natural framework for a unified description of brst cohomology both for lagrangian and hamiltonian systems finally it is discussed how one can in various typical situations replace invariance under an infinite dimensional gauge group by invariance under a suitable finite dimensional lie group this is a necessary prerequisite for handling brst cohomology for such systems within a completely finite dimensional setting

gunther s formalism k symplectic formalism in classical field theory skinner rusk approach and the evolution operator the first aim of this paper is to extend the skinner rusk formalism on classical mechanics for first order field theories the second is to generalize the definition and properties of the evolution k operator on classical mechanics for first order field theories using in both cases gunther s formalism k symplectic formalism

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

constraint field systems in multimomentum canonical variables to describe constraint field models we apply the multimomentum hamiltonian formalism where momenta correspond to derivatives of fields with respect to all world coordinates not only time if a lagrangian density is degenerate the euler lagrange equations are underdetermined and need additional gauge type conditions which remain elusive in general one gets these conditions automatically as a part of the hamilton equations but must consider a family of hamiltonian forms associated with the same lagrangian density in order to exhaust solutions of the euler lagrange equations the case of degenerate quadratic and affine lagrangian densities is elaborated as a result we get the universal procedure of describing constraint field systems

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

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

on the similarity solution of the fragmentation equation an analysis of the similarity solution of the linear homogeneous fragmentation equation is given for a general volume conserving daughter fragment distribution for the special case of a polynomial daughter distribution of degree p an exact basic similarity solution for the time evolution of the particle volume distribution is derived the solution is proportional to the meijer g function which may be represented as a linear combination of generalized hypergeometric functions the properties of generalized hypergeometric functions and g functions are given in the special function literature and include the continuation of the solution to large values of the similarity variable which is given here for special cases

hamiltonian systems on k cosymplectic manifolds the hamiltonian framework on symplectic and cosymplectic manifolds is extended in order to consider classical field theories to do this the notion of k cosymplectic manifold is introduced and a suitable hamiltonian formalism is developed so that the field equations for scalar and vector hamiltonian functions are derived

hamilton jacobi theory for constrained systems we extend the hamilton jacobi formulation to constrained dynamical systems the discussion covers both the case of first class constraints alone and that of first and second class constraints combined the hamilton dirac equations are recovered as characteristic of the system of partial differential equations satisfied by the hamilton jacobi function

lexicographic probabilities and choice under uncertainty conventional bayesian theory of choice under uncertainty subjective expected utility theory fails to satisfy the properties of admissibility and existence of well defined conditional probabilities weakly dominated acts may be chosen and the usual definition of conditional probabilities applies only to nonnull events this paper develops a non archimedean variant of subjective expected utility where decisionmakers have lexicographic beliefs this generalization can be made to satisfy admissibility and yield well defined conditional probabilities and at the same time allow for null events copyright 1991 by the econometric society

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

nk almost tangent structures and the hamiltonization of higher order field theories in this paper a new interpretation of the hamiltonization of higher order lagrangian field theories is given introducing a new class of tensor fields of type 1 1 that extends almost tangent structures to the tangent bundle of nk velocities

the polysymplectic hamiltonian formalism in field theory and calculus of variations i the local case on presente un formalisme hamiltonien geometrique invariant pour des problemes variationnels a integrale multiple et des theories de champ le formalisme est base sur la notion de forme polysymplectique qui est une generalisation a valeur vectorielle des formes symplectiques on etudie des equations hamiltoniennes des transformations canoniques des systemes de lagrange des symetries des applications en pulsion de theorie du champ et une classification des systemes de theorie du champ g homogenes sur une generalisation des orbites coadjointes

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

distributional fixed point equations for island nucleation in one dimension the inverse problem the self consistency of the distributional fixed point equation dfpe approach to understanding the statistical properties of island nucleation and growth during submonolayer deposition is explored we perform kinetic monte carlo simulations in which point islands nucleate on a one dimensional lattice during submonolyer deposition with critical island size i and examine the evolution of the inter island gaps as they are fragmented by new island nucleation the dfpe couples the fragmentation probability distribution within the gaps to the consequent gap size distribution gsd and we find a good fit between the dfpe solutions and the observed gsds for i 0 1 2 3 furthermore we develop numerical methods to address the inverse problem namely the problem of obtaining the gap fragmentation probability from the observed gsd and again find good self consistency in the approach this has consequences for its application to experimental situations where only the gsd is observed and where the growth rules embodied in the fragmentation process must be deduced

k symplectic structures the basic properties of k symplectic structures are introduced and developed in analogy with the well known symplectic differential geometry examples of such structures related to k symplectic lie algebras are given

wild hyperbolic sets yet no chance for the coexistence of infinitely many klus simple newhouse attracting sets the phenomenon of the coexistence of infinitely many sinks for two dimensional dissipative diffeomorphisms is a result due to newhouse ne1 ne2 in fact for each parameter value at which a homoclinic tangency is formed nondegenerately there exist intervals in the parameter space containing dense sets of parameter values for which there are infinitely many coexisting sinks robinson r the structure of the sinks constructed by newhouse is limited simple newhouse parameter values are values at which there are infinitely many sinks having some special well defined property concerning the structure a result due to tedeschini lalli and yorke ty says that the lebesgue measure of the set of simple newhouse parameter values is zero when the tangencies are due to the standard affine horseshoe map it is argued in ty and pr that a more general derivation of this measure zero result would be desirable the main result of this paper is that the lebesgue measure of the set of klus simple parameter values including the simple newhouse parameter values is zero for saddle hyperbolic basic sets forming tangencies

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

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

the limits of ex post implementation the sensitivity of bayesian implementation to agents beliefs about others suggests the use of more robust notions of implementation such as ex post implementation which requires that each agent s strategy be optimal for every possible realization of the types of other agents we show that the only deterministic social choice functions that are ex post implementable in generic mechanism design frameworks with multidimensional signals interdependent valuations and transferable utilities are constant functions in other words deterministic ex post implementation requires that the same alternative must be chosen irrespective of agents signals the proof shows that ex post implementability of a nontrivial deterministic social choice function implies that certain rates of information substitution coincide for all agents this condition amounts to a system of differential equations that are not satisfied by generic valuation functions

semi algebraic functions have small subdifferentials we prove that the subdifferential of any semi algebraic extended real valued function on mathbf r n has n dimensional graph we discuss consequences for generic semi algebraic optimization problems

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

an alternative approach to the cartan form in lagrangian field theories an operator analogous to the almost tangent structure on a tangent bundle is defined on jet bundles of a fibred manifold this operator is used to construct a cartan form the construction is unique for first order lagrangians and is also unique when restricted to higher order mechanics

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

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

distributional fixed point equations for island nucleation in one dimension a retrospective approach for capture zone scaling the distributions of inter island gaps and captures zones for islands nucleated on a one dimensional substrate during submonolayer deposition are considered using a novel retrospective view this provides an alternative perspective on why scaling occurs in this continuously evolving system distributional fixed point equations for the gaps are derived both with and without a mean field approximation for nearest neighbour gap size correlation solutions to the equations show that correct consideration of fragmentation bias justifies the mean field approach which can be extended to provide closed from equations for the capture zones our results compare favourably to monte carlo data for both point and extended islands using a range of critical island size i 0 1 2 3 we also find satisfactory agreement with theoretical models based on more traditional fragmentation theory approaches

jet bundle formulation of infinite dimensional port hamiltonian systems using differential operators we consider infinite dimensional port hamiltonian systems described on jet bundles based on a power balance relation we introduce the port hamiltonian system representation using differential operators regarding the structural mapping the dissipation mapping and the input mapping in contrast to the well known representation on the basis of the underlying stokes dirac structure our approach is not necessarily based on using energy variables which leads to a different port hamiltonian representation of the analyzed partial differential equations the presented constructions will be specialized to mechanical systems to which class also the presented examples belong

on casimir functionals for infinite dimensional port hamiltonian control systems we consider infinite dimensional port hamiltonian systems with respect to control issues in contrast to the well established representation relying on stokes dirac structures that are based on skew adjoint differential operators and the use of energy variables we employ a different port hamiltonian framework based on this system representation conditions for casimir functionals will be derived where in this context the variational derivative plays an extraordinary role furthermore the coupling of finite and infinite dimensional systems will be analyzed in the spirit of the control by interconnection problem

capture zone areas in submonolayer nucleation effects of dimensionality and short range interactions simulation results for the asymptotic scaled capture zone distribution czd for the case of irreversible nucleation and growth of point islands are presented for substrate dimension d 1 2 3 and 4 and compared with a recent conjecture based on the wigner distribution poor agreement is found between the predicted wigner distributions and the asymptotic czd in the limit of infinite df corresponding to the ratio of monomer hopping rate d to deposition rate f our results also indicate that for d 2 and 3 the asymptotic czd for point islands is independent of model details and dimension however for d 1 and d 4 the resulting distribution is significantly more sharply peaked we also find that in contrast to the island size distribution for which mean field like behavior is observed in d 3 and above the asymptotic czd is significantly broadened by fluctuations even in d 4

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

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

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