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

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

an asymptotic theory for dynamic response of anisotropic inhomogeneous and laminated plates abstract we develop an asymptotic theory for dynamic analysis of anisotropic inhomogeneous plates within the framework of three dimensional elasticity the inhomogeneities are considered to be in the thickness direction and the laminated plates belong to an important class of this type of inhomogeneous plates through nondimensionalization and introduction of the multiple time scales in the formulation we obtain a uniform expansion of the field variables in even powers of a small plate parameter the expansion yields an asymptotic solution valid regardless of the time span whereas a straightforward expansion fails to produce convergent results we show by successive integration that the equations for the asymptotic solution are of the same form as those in the classical laminated plate theory clt while the asymptotic solution is no more difficult than the clt solution it is capable of yielding displacements and all the stress components in a consistent and systematic manner modifications to the lower order solutions are made by eliminating the secular terms in the equations according to the method of multiple scales the basic theory is illustrated by determining the free vibration characteristics of a symmetric cross ply laminated plate

the lie poisson structure of integrable classical non linear sigma models the canonical structure of classical non linear sigma models on riemannian symmetric spaces which constitute the most general class of classical non linear sigma models known to be integrable is shown to be governed by a fundamental poisson bracket relation that fits into ther s matrix formalism for non ultralocal integrable models first discussed by maillet the matricesr ands are computed explicitly and being field dependent satisfy fundamental poisson bracket relations of their own which can be expressed in terms of a new numerical matrixc it is proposed that all these poisson brackets taken together are representation conditions for a new kind of algebra which for this class of models replaces the classical yang baxter algebra governing the canonical structure of ultralocal models the poisson brackets for the transition matrices are also computed and the notorious regularization problem associated with the definition of the poisson brackets for the monodromy matrices is discussed

asymptotic theory for laminated piezoelectric circular cylindrical shells an asymptotic theory is presented for laminated piezoelectric circular cylindrical shells under electromechanical loads the three dimensional coupled electromechanical problem reduces to a hierarchy of two dimensional equations which can be solved systematically the proposed theory is illustrated by considering two example problems numerical results show excellent agreement with an available exact solution and also provide some suggestions in establishing new two dimensional piezoelectric shell models

new convergence estimates for multigrid algorithms in this paper new convergence estimates are proved for both symmetric and nonsymmetric multigrid algorithms applied to symmetric positive definite problems our theory relates the convergence of multigrid algorithms to a regularity and approximation parameter cap alpha epsilon 0 1 and the number of relaxations m we show that for the symmetric and nonsymmetric nu cycles the multigrid iteration converges for any positive m at a rate which deteriorates no worse than 1 cj sup 1 sup cap alpha sup sup cap alpha where j is the number of grid levels we then define a generalized nu cycle algorithm which involves exponentially increasing for example doubling the number of smoothings on successively coarser grids we show that the resulting symmetric and nonsymmetric multigrid iterations converge for any cap alpha with rates that are independent of the mesh size the theory is presented in an abstract setting which can be applied to finite element multigrid and finite difference multigrid methods

hamiltonian formulation of the kdv equation we consider the canonical formulation of whitham s variational principle for the kdv equation this lagrangian is degenerate and we have found it necessary to use dirac s theory of constrained systems in constructing the hamiltonian earlier discussions of the hamiltonian structure of the kdv equation were based on various different decompositions of the field which is avoided by this new approach

multigrid algorithms for a vertex centered covolume method for elliptic problems we analyze v cycle multigrid algorithms for a class of perturbed problems whose perturbation in the bilinear form preserves the convergence properties of the multigrid algorithm of the original problem as an application we study the convergence of multigrid algorithms for a covolume method or a vertex centered finite volume element method for variable coefficient elliptic problems on polygonal domains as in standard finite element methods the v cycle algorithm with one pre smoothing converges with a rate independent of the number of levels various types of smoothers including point or line jacobi and gauss seidel relaxation are considered

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

the geometry of the bundle of connections a generalized symplectic structure on the bundle of connections p colon c p rightarrow m of an arbitrary principal g bundle pi colon p rightarrow m is defined by means of a p ast mathrm ad p valued differential 2 form omega 2 on c p which is related to the generalized contact structure on j 1 p the hamiltonian properties of omega 2 are also analyzed

three dimensional asymptotic analysis of multiple electroded piezoelectric laminates theproblemofpiezoelectriclaminateswithspecie edsurfacetractionsandsurfaceandinternalelectricpotentials is studied by writing the governing equations in the state space formulation employing an asymptotic expansion technique and expressing electric displacement jumps across internal electrodes in terms of basic unknowns the three dimensional problem is reduced to a hierarchy of two dimensional equations with the same homogeneous operators for each order different nonhomogeneous terms are only related to the preceding order solution and can be readily determined by recurrence relations moreover for pure elasticity the present e eld equations of the leading order represent the classical thin elastic plate model the proposed formulation is illustrated by considering arectangular piezoelectric plate madeof an orthotropic material and with its edges simplysupported and grounded the convergence of the solution is discussed and the repeated averaging technique for partial sums is used to accelerate the convergenceof the series solution computed results arefound toagree well with available analytical results and new results for electromechanically coupled problems are presented

harmonic maps into lie groups classical solutions of the chiral model on considere deux aspects de la structure algebrique de 7b m l espace des applications harmoniques d un domaine a 2 dimensions simplement connexe dans un groupe de lie reel g r la forme reelle d un groupe complexe g on construit une representation du groupe boucle 7b a s 1 g r sur 7b m correspondant a l algebre de lie de kacmoody des deformations infinitesimales on decrit l action sur 7b m et l action d un sous groupe sur l espace des applications harmoniques dans des gaussmanniens dans une seconde partie on se restreint a une theorie qui s applique seulement quand est un domaine riemannien a 2 dimensions simplement connexe et le groupe est g r u n

euler poincare reduction for discrete field theories in this note we develop a theory of euler poincare reduction for discrete lagrangian field theories we introduce the concept of euler poincare equations for discrete field theories as well as a natural extension of the moser veselov scheme and show that both are equivalent the resulting discrete field equations are interpreted in terms of discrete differential geometry an application to the theory of discrete harmonic mappings is also briefly discussed

reduction of polysymplectic manifolds the aim of this paper is to generalize the classical marsden weinstein reduction procedure for symplectic manifolds to polysymplectic manifolds in order to obtain quotient manifolds which inherit the polysymplectic structure this generalization allows us to reduce polysymplectic hamiltonian systems with symmetries such as those appearing in certain kinds of classical field theories as an application of this technique an analogue to the kirillov kostant souriau theorem for polysymplectic manifolds is obtained and some other mathematical examples are also analyzed our procedure corrects some mistakes and inaccuracies in previous papers gunther 1987 j differ geom 25 23 53 munteanu et al 2004 j math phys 45 1730 51 on this subject

su n and so n invariant chiral fields one and two dimensional subspaces applying the harmonic map ansatz we find new classes of su n and so n chiral fields we write the corresponding one dimensional subgroups of su n and so n chiral fields in terms of one and two harmonic maps and the corresponding two dimensional subgroups in terms of su 2 and so 3 chiral fields in other words we reduce the su n and so n chiral equations to harmonic maps in one and two dimensional spaces

cell centered multigrid for interface problems abstract a multigrid method is presented for cell centered discretizations of elliptic partial differential equations the method works both for smooth and strongly discontinuous coefficients even though in contrast with earlier works the prolongation and restriction operators do not depend on the equation

k symplectic formalism on lie algebroids in this paper we introduce a geometric description of lagrangian and hamiltonian classical field theories on lie algebroids in the framework of k symplectic geometry we discuss the relation between the lagrangian and hamiltonian descriptions through a convenient notion of legendre transformation the theory is a natural generalization of the standard one in addition other interesting examples are studied in particular systems with symmetry and poisson sigma models

on the geometry of spontaneous symmetry breaking given a principal bundle p x with a structure group g and an associated higgs bundle with a standard fiber g h the case of a matter bundle e whose standard fiber admits action only of an exact symmetry subgroup h of g is examined in the presence of a fixed higgs field x matter fields are represented by sections of a matter bundle eh associated with the corresponding reduced subbundle ph of p the totality of matter fields and higgs fields is described by sections of the bundle e which is the composite bundle eh x where eh is the bundle associated with the principal h bundle p the bundle e fails to be associated with a principal bundle to construct a connection e j1e on e the canonical jet bundle morphism j1eh j1 j1e is used

reduced classical field theories k cosymplectic formalism on lie algebroids in this paper we introduce a geometric description of lagrangian and hamiltonian classical field theories on lie algebroids in the framework of k cosymplectic geometry we discuss the relation between lagrangian and hamiltonian descriptions through a convenient notion of legendre transformation the theory is a natural generalization of the standard one in addition other interesting examples are studied mainly on reduction of classical field theories

asymptotic splitting in the three dimensional problem of elasticity for non homogeneous piezoelectric plates a novel asymptotic approach to the theory of non homogeneous anisotropic plates is suggested for the problem of linear static deformations we consider solutions which are slowly varying in the plane of the plate in comparison to the thickness direction a small parameter is introduced in the general equations of the theory of elasticity according to the procedure of asymptotic splitting the principal terms of the series expansion of the solution are determined from the conditions of solvability for the minor terms three dimensional conditions of compatibility make the analysis more efficient and straightforward we obtain the system of equations of classical kirchhoff s plate theory including the balance equations compatibility conditions elastic relations and kinematic relations between the displacements and strain measures subsequent analysis of the edge layer near the contour of the plate is required in order to satisfy the remaining boundary conditions of the three dimensional problem matching of the asymptotic expansions of the solution in the edge layer and inside the domain provides four classical plate boundary conditions additional effects like electromechanical coupling for piezoelectric plates can easily be incorporated into the model due to the modular structure of the analysis the results of the paper constitute a sound basis to the equations of the theory of classical plates with piezoelectric effects and provide a trustworthy algorithm for computation of the stressed state in the three dimensional problem numerical and analytical studies of a sample electromechanical problem demonstrate the asymptotic nature of the present theory

classical field theory on lie algebroids variational aspects the variational formalism for classical field theories is extended to the setting of lie algebroids given a lagrangian function we study the problem of finding critical points of the action functional when we restrict the fields to being morphisms of lie algebroids in addition to the standard case our formalism includes as particular examples the case of systems with symmetry covariant euler poincare and lagrange poincare cases sigma models or chern simons theories

shape control of vibrating simply supported rectangular plates we determine the optimum location of a given rectangular piezoceramic pzt actuator that will require the minimum voltage to annul the deflections of a simply supported rectangular linear elastic plate vibrating near one of its fundamental frequencies keeping the location of the centroid and the shape of the pzt fixed we ascertain the voltage required as a function of the length of its diagonal to nullify the deflections of the plate

an asymptotic variational formulation for dynamic analysis of multilayered anisotropic plates abstract a multilevel variational formulation for dynamic analysis of multilayered anisotropic plates is developed within the framework of three dimensional elasticity by means of asymptotic expansions the hellinger reissner functional for the elastodynamic problem is decomposed into a series of functionals with which a computational model can be constructed in the formulation multiple time scales are introduced so that the secular terms can be eliminated systematically in obtaining a uniform expansion leading to valid asymptotic solution modifications to the approximation of various orders are determined by considering the solvability conditions of the higher order equations the model is adaptive when combined with the finite element method it has many appealing features including that the displacements and transverse stresses may be interpolated independently that the nodal degree of freedom dof at each level is less than that of kirchhoff plates and that the mass and stiffness matrices generated at the leading order level are always used at subsequent levels above all the solution is three dimensional in effect yet requires only two dimensional interpolation the through thickness variations of the field variables are determined analytically with no need of interpolating in the thickness direction

on the bundle of connections and the gauge orbit manifold in yang mills theory in an appropriate mathematical framework we supply a simple proof that the quotienting of the space of connections by the group of gauge transformations in yang mills theory is ac principal fibration the underlying quotient space the gauge orbit space is seen explicitly to be ac manifold modelled on a hilbert space

reduction in principal bundles covariant lagrange poincare equations a general reduction theory for field theoretical lagrangians on principal fiber bundles is presented the reduced variational problem as well as the associated equations are formulated the link between the solutions of the original problem and that of the reduced problem is discussed and an obstruction to the reconstruction of the solutions is isolated the important case of semidirect products is discussed in detail and several concrete examples are presented

symmetry reduction of variational bicomplexes and the principle of symmetric criticality consider a system of differential equations a 0 which is invariant under a lie group g of point transformations acting on the space e of independent and dependent variables by a method due to lie the g invariant solutions of these differential equations are found by solving a reduced system of differential equations a 0 on the space e of invariants of g in this paper we explore the relationship between the g invariant conservation laws and variational principles for the system of equations a 0 and the conservation laws and variational principles for the reduced equations a 0 this problem translates into one of constructing a certain cochain map qx between the g invariant variational bicomplex for the infinite jet space on e and the free variational bicomplex for e we prove that such a cochain map exists locally if and only if the relative lie algebra cohomology condition h f tooo 0 is satisfied where q is the orbit dimension of g r the lie algebra of vector fields on e which generate the infinitesimal action of g and toc o the linear isotropy subalgebra of t at eo as a simple consequence we prove that the vanishing of hq t tooo is the only local obstruction to palais principle of symmetric criticality 1 introduction when a system of partial differential equations is invari ant under a lie group g of point transformations one can construct a reduced system of partial or ordinary differential equations whose local solutions cor respond precisely to the local g invariant solutions of the original system of equations this simple and effective construction pioneered by lie himself is still one of the most powerful and extensively used techniques for finding exact solutions of many nonlinear partial differential equations 4 15 23 in this paper we explore the relationship between the g invariant conservation laws and variational principles for a system of partial differential equations and the con servation laws and variational principles of the reduced system of equations for the g invariant solutions since conservation laws and variational principles for differential equations are naturally represented by certain cohomology classes in the variational bicomplex associated to the given system of partial differential equations 1 22 our problem translates into that of constructing a cochain map between certain variational bicomplexes remarkably and contrary to the usual spirit of symmetry reduction these cochain maps do not always exist even lo cally we are able to completely classify the local obstructions to the existence of such cochain maps

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

multigrid algorithms for nonconforming and mixed methods for nonsymmetric and indefinite problems in this paper we consider multigrid algorithms for nonconforming and mixed finite element methods for nonsymmetric and or indefinite elliptic problems we show that a simple cal v cycle multigrid iteration using conforming coarse grid corrections converges at a uniform rate provided that the coarsest level in the multilevel iteration is sufficiently fine but independent of the number of multigrid levels various types of smoothers for the nonsymmetric and indefinite problems are discussed extensive numerical results are presented

the analysis of elliptic families i metrics and connections on determinant bundles in this paper we construct the quillen metric on the determinant bundle associated with a family of elliptic first order differential operators we also introduce a unitary connection on and calculate its curvature our results will be applied to the case of dirac operators in a forthcoming paper

multigrid algorithm for cell centered finite difference on triangular meshes1this work was partially supported by kosef contract number 971 0106 342 2 1 we consider a multigrid algorithm for the cell centered difference scheme on triangular meshes using a new prolongation operator the energy norm of this prolongation is shown to be less than 2 thus the w cycle is guaranteed to converge numerical experiments show that our operator is better than the trivial injection

a refined asymptotic theory and computational model for multilayered composite plates abstract the asymptotic theory of multilayered composite plates developed recently is refined to embrace the shear deformation theory the derivation is based on application of asymptotic expansion to the hellinger reissner variational functional in which the displacements and transverse stresses are taken to be the functions subject to variation by introducing rotations as auxiliary variables in the asymptotic formulation the shear deformation theory of laminated plates arises naturally as the first order approximation to the three dimensional theory higher order corrections are determined in an adaptive and hierarchic way the computational model is constructed on the basis of the asymptotic expressions of the h r functional in conjunction with the finite element method in which the displacements and transverse stresses may be interpolated independently through successive ntegration the transverse stress degrees of freedom are condensed at the element level resulting in rotations and midplane displacements as basic unknowns in the system equations in the model the lateral boundary conditions are satisfied exactly the edge conditions associated with each level are shown to be similar to the ones in the 2d theory of mindlin s plates which can be easily implemented in the computation the performance of the model is illustrated by example problems

multilayered piezoelectric refined plate theory a two dimensional theory for the analysis of multilayered piezoelectric plates is presented the theory is based on a hybrid approach in which the continuity conditions for both mechanical and electric unknowns at layer interfaces as well as the imposed conditions on the bounding surfaces and at the interfaces are independently satisfied then the piezoelectric boundary value problem is stated using mechanical displacements and electrostatic potential in conjunction with the coupled piezoelectric constitutive law the proposed model is simple to use incorporating only five independent generalized displacements and one or two independent generalized electrostatic potentials as unknowns the number of electric unknowns depends on the number of layers and electric prescribed conditions the accuracy of the proposed theory is assessed through investigation of significant problems for which an exact three dimensional solution is known from heyliger first in dynamics the free vibrations of five layered piezoelectric plates and second in statics for three layered plates with an imposed force density results obtained with our model compare very well with the exact three dimensional theory

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

an efficient coupled layerwise theory for dynamic analysis of piezoelectric composite beams an efficient new coupled one dimensional model is developed for the dynamics of piezoelectric composite beams the model combines third order zigzag approximation for the displacement with layerwise approximation of the electric field as piecewise linear for sublayers by enforcing the conditions of zero transverse shear stress at the top and bottom and its continuity at layer interfaces the displacement field is expressed in terms of three primary displacement variables and potentials the governing coupled equations of stress and charge equilibrium and boundary conditions are derived from hamilton s principle analytical solutions are obtained for free vibrations and forced response under harmonic load for simply supported hybrid beams and the results are compared with the exact three dimensional solution and uncoupled first order shear deformation theory solution the present results show significant improvement over the first order solution and agree very well with the exact solution for both thin and thick hybrid beams the results demonstrate the capability of the developed theory to adequately model open and closed circuit electric boundary conditions to accurately predict their influence on the response

a theoretical analysis of piezoelectric composite laminate with larger amplitude deflection effect part ii hermite differential quadrature method and application incorporating the effects of larger amplitude deflection and electro elastical properties of piezoelectric lamina the hamilton s variation principle was used to deduce the fundamental formulations of smart anisotropic composite plate in part i in terms of reddy s simple higher order theory in order to solve the five highly coupled nonlinear partial differential equations with complicated overlapping boundary conditions a novel numerical method hermite differential quadrature hdq method was developed to implement the differential equations with complicated overlapping boundary conditions based on the presently developed hdq method any orders derivatives of the unknown functions or any boundary conditions can be point collocation based discretized by a set of point values along x and y direction then a system of complete algebraic nonlinear equations can be constructed to calculate out the final point values of the mid plane displacements by using the governing equations and relative boundary conditions with hdq method finally some detailed numerical examples for the anisotropic piezoelectric composite laminate with the distributed poling directions of piezoelectric layer and fiber orientations of composite layers were studied to validate the developed theoretical analysis model and hdq numerical method