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

an online method for interpolating linear parametric reduced order models a two step online method is proposed for interpolating projection based linear parametric reduced order models roms in order to construct a new rom for a new set of parameter values the first step of this method transforms each precomputed rom into a consistent set of generalized coordinates the second step interpolates the associated linear operators on their appropriate matrix manifold real time performance is achieved by precomputing inner products between the reduced order bases underlying the precomputed roms the proposed method is illustrated by applications in mechanical and aeronautical engineering in particular its robustness is demonstrated by its ability to handle the case where the sampled parameter set values exhibit a mode veering phenomenon

nonlinear manifold learning for meshfree finite deformation thin shell analysis calculations on general point set surfaces are attractive because of their flexibility and simplicity in the preprocessing but present important challenges the absence of a mesh makes it nontrivial to decide if two neighboring points in the three dimensional embedding are nearby or rather far apart on the manifold furthermore the topology of surfaces is generally not that of an open two dimensional set ruling out global parametrizations we propose a general and simple numerical method analogous to the mathematical theory of manifolds in which the point set surface is described by a set of overlapping charts forming a complete atlas we proceed in four steps 1 partitioning of the node set into subregions of trivial topology 2 automatic detection of the geometric structure of the surface patches by nonlinear dimensionality reduction methods 3 parametrization of the surface using smooth meshfree here maximum entropy approximants and 4 gluing together the patch representations by means of a partition of unity each patch may be viewed as a meshfree macro element we exemplify the generality flexibility and accuracy of the proposed approach by numerically approximating the geometrically nonlinear kirchhoff love theory of thin shells we analyze standard benchmark tests as well as point set surfaces of complex geometry and topology

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

a receding contact plane problem between a functionally graded layer and a homogeneous substrate abstract in this paper we consider the plane problem of a frictionless receding contact between an elastic functionally graded layer and a homogeneous half space when the two bodies are pressed together the graded layer is modeled as a nonhomogeneous medium with an isotropic stress strain law and over a certain segment of its top surface is subjected to normal tractions while the rest of this surface is free of tractions since the contact between the two bodies is assumed to be frictionless then only compressive normal tractions can be transmitted in the contact area using integral transforms the plane elasticity equations are converted analytically into a singular integral equation in which the unknowns are the contact pressure and the receding contact half length the global equilibrium condition of the layer is supplemented to solve the problem the singular integral equation is solved numerically using chebychev polynomials and an iterative scheme is employed to obtain the correct receding contact half length that satisfies the global equilibrium condition the main objective of the paper is to study the effect of the material nonhomogeneity parameter and the thickness of the graded layer on the contact pressure and on the length of the receding contact

variational time integrators for finite dimensional thermo elasto dynamics without heat conduction this paper focuses on the formulation of variational integrators vis for finite dimensional thermo elastic systems without heat conduction the dynamics of these systems happen to have a hamiltonian structure after thermal displacements are introduced it is then possible to formulate integrators by taking advantage of standard methods in vi the class of integrators we construct have some remarkable features a they are symplectic b they exactly conserve the entropy of the system or in other words they exactly satisfy the second law of the thermodynamics for reversible adiabatic processes c they nearly exactly conserve the value of the energy for very long times and d they exactly conserve linear and angular momentum we first describe how to adapt any vi for the classical mechanical systems to integrate adiabatic thermo elastic ones and then formulate three new types of integrators the first class based on a generalized trapezoidal rule gives rise to two first order explicit integrators and a second order implicit one by composing then the two first order integrators we construct a second order explicit one finally we formulate a fourth order implicit integrator which is a symplectic partitioned runge kutta method the performance of these new algorithms is showcased through numerical examples copyright 2011 john wiley sons ltd

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

frictional contact problem for a rigid cylindrical stamp and an elastic layer resting on a half plane abstract the frictional contact problem for a layer resting on a homogeneous half plane is handled using linear elasticity theory in this study the layer is in contact with a rigid cylindrical stamp that is on the layer and applies a concentrated force in the normal and tangential directions friction between the component couples of layer stamp and layer half plane is taken into account the problem is reduced to a system of singular integral equations in which the contact pressures and the contact areas are the unknowns and it is treated using fourier transforms and the boundary conditions for the problem the system of singular integral equations is solved numerically using the gauss jacobi integration formula with equilibrium and consistency conditions numerical results for the contact pressures and the contact areas are given as a solution for both the frictional and the frictionless cases this work is the first study that investigates the effect of friction on the receding contact problem of a layer and a half plane with two contact areas

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

solving singular integral equations using gaussian quadrature and overdetermined system abstract gauss chebyshev quadrature and collocation at the zeros of the chebyshev polynomial of the first kind t n x and second kind u n x leads to an overdetermined system of linear algebraic equations the size of the coefficient matrix for the overdetermined system depends on the degrees of chebyshev polynomials used we show that we can get more accurate solution using t 4 n 4 x than other t n x the regularization method using generalized singular value decomposition is described and compared to gauss newton method for solving the overdetermined system of equations computational tests show that gsvd with an appropriate choice of regularization parameter gives better solution in solving singular integral equations

group actions on c algebras 3 cocycles and quantum field theory we study group extensions where acts on a c algebraa given a twisted covariant representation v of the paira we construct 3 cocycles on with values in the centre of the group generated byv these 3 cocycles are obstructions to the existence of an extension of byv which acts ona compatibly with the main theorems of the paper introduce a subsidiary invariant which classifies actions of onv and in terms of which a necessary and sufficient condition for the the cohomology class of the 3 cocycle to be non trivial may be formulated examples are provided which show how non trivial 3 cocycles may be realised the framework we choose to exhibit these essentially mathematical results is influenced by anomalous gauge field theories we show how to interpret our results in that setting in two ways one motivated by an algebraic approach to constrained dynamics and the other by the descent equation approach to constructing cocycles on gauge groups in order to make comparisons with the usual approach to cohomology in gauge theory we conclude with a lie algebra version of the invariant and the 3 cocycle

solving singular integral equations using gaussian quadrature and overdetermined systems abstract for singular integral equations of the cauchy type on an open interval a bounded solution exists only if an additional condition is satisfied in direct methods of solution when the integrals are replaced by quadratures it is not obvious that the resulting discrete system of equations will be consistent it is shown that for gauss chebyshev quadrature the overdetermined system of equations is consistent if and only if the discrete analog of the compatibility condition is satisfied the behavior of solutions to inconsistent systems is examined

dimensional model reduction in non linear finite element dynamics of solids and structures a general approach to the dimensional reduction of non linear finite element models of solid dynamics is presented for the newmark implicit time discretization the computationally most expensive phase is the repeated solution of the system of linear equations for displacement increments to deal with this it is shown how the problem can be formulated in an approximation ritz basis of much smaller dimension similarly the explicit newmark algorithm can be also written in a reduced dimension basis and the computation time savings in that case follow from an increase in the stable time step length in addition the empirical eigenvectors are proposed as the basis in which to expand the incremental problem this basis achieves approximation optimality by using computational data for the response of the full model in time to construct a reduced basis which reproduces the full system in a statistical sense because of this global time viewpoint the basis need not be updated as with reduced bases computed from a linearization of the full finite element model if the dynamics of a finite element model is expressed in terms of a small number of basis vectors the asymptotic cost of the solution with the reduced model is lowered and optimal scalability of the computational algorithm with the size of the model is achieved at the same time numerical experiments indicate that by using reduced models substantial savings can be achieved even in the pre asymptotic range furthermore the algorithm parallelizes very efficiently the method we present is expected to become a useful tool in applications requiring a large number of repeated non linear solid dynamics simulations such as convergence studies design optimization and design of controllers of mechanical systems

involution and constrained dynamics i the dirac approach we study the theory of systems with constraints from the point of view of the formal theory of partial differential equations for finite dimensional systems we show that the dirac algorithm completes the equations of motion to an involutive system we discuss the implications of this identification for field theories and argue that the involution analysis is more general and flexible than the dirac approach we also derive intrinsic expressions for the number of degrees of freedom

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

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

optimal prediction of underresolved dynamics abstract a method is presented for computing the average solution of problems that are too complicated for adequate resolution but where information about the statistics of the solution is available the method involves computing average derivatives by interpolation based on linear regression and an updating of a measure constrained by the available crude information examples are given

the generalized cattaneo partial slip plane contact problem i theory the cattaneo problem is considered for a general plane contact between elastically similar materials i e a monotonically increasing tangential load starting from zero with normal loading held fixed instead of the classical argument on the displacement field in the stick zone of cattaneo solution we attack the problem implicitly from the governing integral equations in the stick zones after discussing and solving the full stick case we move to the more realistic for finite friction case of partial slip we show that upon isolating the effect of full sliding the equalities and inequalities governing the corrective solution for the corrective shearing tractions in the stick zone are exactly the same as those governing the solution of the normal contact problem with a lower load but the same rotation as the actual one this analogy permits us to deduce several general properties and gives a general procedures for solving partial slip cattaneo problems as frictionless normal indentation ones therefore the general solutions for single multiple and periodic contacts is given a comprehensive set of explicit results is given in the part ii of the paper

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

a new look at degenerate lagrangian dynamics from the viewpoint of almost product structures singular lagrangian systems are studied in the framework of almost product structures the choice of an appropriate almost product structure permits us to obtain the dynamics the relationship with the dirac bracket is also elucidated

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

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

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 manifold learning based reduced order model for springback shape characterization and optimization in sheet metal forming abstract the parameters of a stamping process include the geometry of the tools the shape of the initial sheet blank the material constitutive law and the process parameters when designing the overall process one has to also take into account the springback effect that appears when the tools are removed and additional surfaces are cut off the goal then is to obtain a final shape as close as possible to the desired shape while satisfying the admissibility constraints on the variable parameters as well as the feasibility constraints frequently expressed in the form of forming limit diagrams in the present paper we represent the post springback shape by a level set function then rather than rely on arbitrarily selected case dependent measurement locations as in the numisheet benchmark problems we build a reduced order shape space where this level set evolves by extending our recent shape manifold approach to the problem of springback assessment for 3d shapes next we propose an optimization algorithm designed to minimize the gap between the post springback and the desired final shapes the required level set functions are generated from a corresponding set of springback shapes predicted by finite element simulations using our approach we determine the minimal number of parameters needed in order to uniquely characterize the final formed shape regardless of complexity finally we demonstrate the approach using an industrial test case springback assessment of the deep drawing operation of an automotive strut tower

local maximum entropy approximation schemes a seamless bridge between finite elements and meshfree methods we present a one parameter family of approximation schemes which we refer to as local maximum entropy approximation schemes that bridges continuously two important limits delaunay triangulation and maximum entropy max ent statistical inference local max ent approximation schemes represent a compromise in the sense of pareto optimality between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width local max ent approximation schemes are entirely defined by the node set and the domain of analysis and the shape functions are positive interpolate affine functions exactly and have a weak kronecker delta property at the boundary local max ent approximation may be regarded as a regularization or thermalization of delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation local max ent approximation schemes can be taken as a convenient basis for the numerical solution of pdes in the style of meshfree galerkin methods in test cases characterized by smooth solutions we find that the accuracy of local max ent approximation schemes is vastly superior to that of finite elements

reduction of lie jordan banach algebras and quantum states in this paper it is shown that the concept of dynamical correspondence for jordan banach algebras is equivalent to a lie structure compatible with the jordan one then a theory of reduction of lie jordan banach algebras in the presence of quantum constraints is presented and compared to the standard reduction of c algebras of observables of a quantum system the space of states of the reduced lie jordan banach algebra is characterized in terms of dirac states on the physical algebra of observables and its gns representations described in terms of states on the unreduced algebra

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

the quantum theory of second class constraints kinematics the problem of second class quantum constraints is here set up in the context ofc algebras utilizing the connection with state conditions as given by the heuristic quantization rules that is a constraint set is said to be first class if all its members can satisfy the same state condition and second class otherwise several heuristic models are examined and they all agree with this definition given then a second class constraint set we separate out its first class part as all those constraints which are compatible with the others and we propose an algebraic construction for imposition of the constraints this construction reduces to the normal one when the constraints are first class moreover the physical automorphisms assumed as conserving the constraints will also respect this construction the final physical algebra obtained is free of constraints gauge invariant unital and with the right choice simple thisc algebra also contains a factor algebra of the usual observables i e the commutator algebra of the constraints the general theory is applied to two examples the elimination of a canonical pair from a boson field theory as in the two dimensional anomalous chiral schwinger model of rajaraman 14 and the imposition of quadratic second class constraints on a linear boson field theory

nonlinear manifold learning for model reduction in finite elastodynamics abstract model reduction in computational mechanics is generally addressed with linear dimensionality reduction methods such as principal components analysis pca hypothesizing that in many applications of interest the essential dynamics evolve on a nonlinear manifold we explore here reduced order modeling based on nonlinear dimensionality reduction methods such methods are gaining popularity in diverse fields of science and technology such as machine perception or molecular simulation we consider finite deformation elastodynamics as a model problem and identify the manifold where the dynamics essentially take place the slow manifold by nonlinear dimensionality reduction methods applied to a database of snapshots contrary to linear dimensionality reduction the smooth parametrization of the slow manifold needs special techniques and we use local maximum entropy approximants we then formulate the lagrangian mechanics on these data based generalized coordinates and develop variational time integrators our proof of concept example shows that a few nonlinear collective variables provide similar accuracy to tens of pca modes suggesting that the proposed method may be very attractive in control or optimization applications furthermore the reduced number of variables brings insight into the mechanics of the system under scrutiny our simulations also highlight the need of modeling the net effect of the disregarded degrees of freedom on the reduced dynamics at long times

on the optimum support size in meshfree methods a variational adaptivity approach with maximum entropy approximants we present a method for the automatic adaption of the support size of meshfree basis functions in the context of the numerical approximation of boundary value problems stemming from a minimum principle the method is based on a variational approach and the central idea is that the variational principle selects both the discretized physical fields and the discretization parameters here those defining the support size of each basis function we consider local maximum entropy approximation schemes which exhibit smooth basis functions with respect to both space and the discretization parameters the node location and the locality parameters we illustrate by the poisson linear and non linear elasticity problems the effectivity of the method which produces very accurate solutions with very coarse discretizations and finds unexpected patterns of the support size of the shape functions copyright 2009 john wiley sons ltd

nonlinear model reduction via discrete empirical interpolation a dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition pod method for constructing reduced order models for time dependent and or parametrized nonlinear partial differential equations pdes in the presence of a general nonlinearity the standard pod galerkin technique reduces dimension in the sense that far fewer variables are present but the complexity of evaluating the nonlinear term remains that of the original problem the original empirical interpolation method eim is a modification of pod that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by pod we propose a discrete empirical interpolation method deim a variant that is suitable for reducing the dimension of systems of ordinary differential equations odes of a certain type as presented here it is applicable to odes arising from finite difference discretization of time dependent pdes and or parametrically dependent steady state problems however the approach extends to arbitrary systems of nonlinear odes with minor modification our contribution is a greatly simplified description of the eim in a finite dimensional setting that possesses an error bound on the quality of approximation an application of deim to a finite difference discretization of the one dimensional fitzhugh nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long time integration that fully captures nonlinear limit cycle behavior we also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results

maximum entropy methods for time harmonic acoustics abstract this paper explores the application of maximum entropy methods max ent to time harmonic acoustic problems max ent basis functions are mesh free approximants that are constructed observing an equivalence between basis functions and discrete probability distributions and applying jaynes s maximum entropy principle they are c continuous and therefore they are particularly suited for the resolution of helmholtz problems where classical finite element methods show a poor accuracy in the high frequency region in addition it was recently shown that max ent approximants can be blended with isogeometric basis functions on the boundary of the domain this preserves the correct representation of the boundary like in isogeometric analysis with the advantage that the discretization of the interior of the domain is straightforward in this paper the max ent mathematical formulation is reviewed and then some numerical applications are studied including a 2d car cavity geometry defined by b spline curves in all cases if the same nodal discretization is used finite elements results are significantly improved

a receding contact axisymmetric problem between a functionally graded layer and a homogeneous substrate abstract in this paper we consider the axisymmetric problem of a frictionless receding contact between an elastic functionally graded layer and a homogeneous half space when the two bodies are pressed together the graded layer is modeled as a nonhomogeneous medium with an isotropic stress strain law and is subjected over a part of its top surface to normal tractions while the rest of it is free of tractions since the contact between the two bodies is assumed to be frictionless then only compressive normal tractions can be transmitted in the contact area using hankel transform the axisymmetric elasticity equations are converted analytically into a singular integral equation in which the unknowns are the contact pressure and the receding contact radius the global equilibrium condition of the layer is supplemented to solve the problem the singular integral equation is solved numerically using orthogonal chebychev polynomials and an iterative scheme is employed to obtain the correct receding contact length that satisfies the global equilibrium condition the main objective of the paper is to study the effect of the material nonhomogeneity parameter and the thickness of the graded layer on the contact pressure and on the length of the receding contact

the hamiltonian structure of yang mills theories and instantons ii abstract the formalism of constraints reviewed in paper i is applied to yang mills theory to determine the physical phase space this turns out to be the cotangent bundle of orbit space the space of gauge inequivalent potentials self dual configurations are not hamiltonian with respect to the symplectic structure inherited from the general system

a frictional receding contact plane problem between a functionally graded layer and a homogeneous substrate abstract this paper investigates the plane problem of a frictional receding contact formed between an elastic functionally graded layer and a homogeneous half space when they are pressed against each other the graded layer is assumed to be an isotropic nonhomogeneous medium with an exponentially varying shear modulus and a constant poisson s ratio a segment of the top surface of the graded layer is subject to both normal and tangential traction while rest of the surface is devoid of traction the entire contact zone thus formed between the layer and the homogeneous medium can transmit both normal and tangential traction it is assumed that the contact region is under sliding contact conditions with the coulomb s law used to relate the tangential traction to the normal component employing fourier integral transforms and applying the necessary boundary conditions the plane elasticity equations are reduced to a singular integral equation in which the unknowns are the contact pressure and the receding contact lengths ensuring mechanical equilibrium is an indispensable requirement warranted by the physics of the problem and therefore the global force and moment equilibrium conditions for the layer are supplemented to solve the problem the gauss chebyshev quadrature collocation method is adopted to convert the singular integral equation to a set of overdetermined algebraic equations this system is solved using a least squares method coupled with a novel iterative procedure to ensure that the force and moment equilibrium conditions are satisfied simultaneously the main objective of this paper is to study the effect of friction coefficient and nonhomogeneity factor on the contact pressure distribution and the size of the contact region

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

variational time integrators the purpose of this paper is to review and further develop the subject of variational integration algorithms as it applies to mechanical systems of engineering interest in particular the conservation properties of both synchronous and asynchronous variational integrators avis are discussed in detail we present selected numerical examples which demonstrate the excellent accuracy conservation and convergence characteristics of avis in these tests avis are found to result in substantial speed ups at equal accuracy relative to explicit newmark a mathematical proof of convergence of the avis is also presented in this paper finally we develop the subject of horizontal variations and configurational forces in discrete dynamics this theory leads to exact path independent characterizations of the configurational forces acting on discrete systems notable examples are the configurational forces acting on material nodes in a finite element discretisation and the j integral at the tip of a crack in a finite element mesh

algebraic quantization of systems with a gauge degeneracy systems with a gauge degeneracy are characterized either by supplementary conditions or by a set of generators of gauge transformations or by a set of constraints deriving from dirac s canonical constraint method these constraints can be expressed either as conditions on the field algebra or on the states on in ac algebra framework we show that the state conditions give rise to a factor algebra of a subalgebra of the field algebra this factor algebra is free of state conditions in this formulation we show also that the algebraic conditions can be treated in the same way as the state conditions the connection between states on and states on is investigated further within this framework as is also the set of transformations which are compatible with the set of constraints it is also shown that not every set of constraints can give rise to a nontrivial system finally as an example the abstract theory is applied to the electromagnetic field and this treatment can be generalized to all systems of bosons with linear constraints the question of dynamics is not 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

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

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