the hamiltonian structure of general relativistic perfect fluids we show that the evolution equations for a perfect fluid coupled to general relativity in a general lapse and shift are hamiltonian relative to a certain poisson structure for the fluid variables a lie poisson structure associated to the dual of a semi direct product lie algebra is used while the bracket for the gravitational variables has the usual canonical symplectic structure the evolution is governed by a hamiltonian which is equivalent to that obtained from a canonical analysis the relationship of our hamiltonian structure with other approaches in the literature such as clebsch potentials lagrangian to eulerian transformations and its use in clarifying linearization stability are discussed

the hamiltonian structure for dynamic free boundary problems hamiltonian structures for 2 or 3 dimensional incompressible flows with a free boundary are determined which generalize a previous structure of zakharov for irrotational flow our poisson bracket is determined using the method of arnold namely reduction from canonical variables in the lagrangian material description using this bracket the hamiltonian form for the equations of a liquid drop with a free boundary having surface tension is demonstrated the structure of the bracket in terms of a reduced cotangent bundle of a principal bundle is explained in the case of two dimensional flows the vorticity bracket is determined and the generalized enstrophy is shown to be a casimir function this investigation also clears up some confusion in the literature concerning the vorticity bracket even for fixed boundary flows

spatially periodic instantons nahm transform and moduli this paper establishes that the nahm transform sending spatially periodic instantons instantons on the product of the real line and a three torus to singular monopoles on the dual three torus is indeed a bijection as suggested by the heuristic in the process we show how the nahm transform intertwines to a fourier mukai transform via kobayashi hitchin correspondences we also prove existence and non existence results

richardson number criterion for the nonlinear stability of three dimensional stratified flow with use of a method of arnol d we derive the necessary and sufficient conditions for the formal stability of a parallel shear flow in a three dimensional stratified fluid when the local richardson number defined with respect to density variations is everywhere greater than unity the equilibrium is formally stable under nonlinear pertrubations the essential physical content of the nonlinear stability result is that the total energy acts as a potential well for deformations of the fluid across constant density surfaces this well is required to have definite curvature to assure stability under these deformations

on the atiyah drinfeld hitchin manin construction for self dual gauge fields the vector spacesa b c in terms of which the general construction due to atiyah drinfeld hitchin and manin for self dual gauge fields defined over some region of euclidean space is phrased are shown to be expressible in terms of the spaces spanned by the solutions of certain linear covariant differential equations depending on the gauge field the corresponding linear maps betweena andb b andc are given with the properties required by adhm and the results then necessary to verify the construction informally proved the local problems associated with assuming the gauge field to obey the self duality equations are separated from the global problems of assuring the required boundary conditions for a particular solution with suitable global conditionsc is shown to be the dual ofa and a natural scalar product defined onb so as to reconstruct the gauge field in the standard form given by the construction a discussion is given of the requirements entailed by the condition of a symmetry on the gauge field and the relation to the usual cohomological treatment is outlined in an appendix

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

the classical shallow water equations symplectic geometry the classical shallow water equations express the change with time of the height h and the velocity of a 1 dimensional fluid t x h x 0 h x h x 0 they possess an infinite number of integrals of motion due to benney 1973 and can be written in hamiltonian form relative to a symplectic structure introduced by manin 1978 the present paper deals with their complete integrability up to the advent of shocks this is proved in the small under an extra assumption satisfied by most height velocity pairs that hh only at isolated points

comment on hamiltonian splitting for the vlasov maxwell equations the paper 1 by crouseilles einkemmer and faou used an incorrect poisson bracket for the vlasov maxwell equations if the correct poisson bracket is used the solution of one of the subsystems cannot be computed exactly in general as a result one cannot construct a symplectic scheme for the vlasov maxwell equations using the splitting hamiltonian method proposed in ref 1

optimal error estimate for energy preserving splitting schemes for maxwell s equations abstract two efficient splitting schemes are developed for 3d maxwell s equations the schemes are energy preserving and unconditionally stable while being implemented explicitly rigorous optimal error estimates are established for the proposed schemes and especially the constant in the error estimates is only o t numerical results confirm the theoretical analysis and numerical comparison with some existing methods shows the good performance of the present schemes

cox rings of cubic surfaces and fano threefolds abstract we determine the cox rings of the minimal resolutions of cubic surfaces with at most rational double points of blow ups of the projective plane at non general configurations of six points and of three dimensional smooth fano varieties of picard numbers one and two

computing cox rings we consider modifications for example blow ups of mori dream spaces and provide algorithms for investigating the effect on the cox ring e g testing finite generation or computing an explicit presentation in terms of generators and relations as a first application we compute the cox rings of all gorenstein log del pezzo surfaces of picard number one moreover we show computationally that all smooth rational surfaces of picard number at most six are mori dream surfaces and we provide explicit presentations of the cox ring for those not admitting a torus action finally we provide the cox rings of projective spaces blown up at a certain special point configurations

hamiltonian structure and darboux theorem for families of generalized lotka volterra systems this work is devoted to the establishment of a poisson structure for a format of equations known as generalized lotka volterra systems these equations which include the classical lotka volterra systems as a particular case have been deeply studied in the literature they have been shown to constitute a whole hierarchy of systems the characterization of which is made in the context of simple algebra our main result is to show that this algebraic structure is completely translatable into the poisson domain important poisson structures features such as the symplectic foliation and the darboux canonical representation rise as a result of rather simple matrix manipulations

frobenius manifold for the dispersionless kadomtsev petviashvili equation we consider a frobenius structure associated with the dispersionless kadomtsev petviashvili equation this is done essentially by applying a continuous analogue of the finite dimensional theory in the space of schwartz functions on the line the potential of the frobenius manifold is found to be a logarithmic energy with quadratic external field following the construction of the principal hierarchy we construct a set of infinitely many commuting flows which extends the classical dkp hierarchy

gauge independent canonical formulation of relativistic plasma theory abstract following an earlier work we derive a gauge independent canonical structure for a fully relativistic multicomponent plasma theory the klimontovich form of the distribution function is used to derive the basic poisson bracket relations for the canonical variables b and e the poisson bracket relations provide an explicit canonical realization of the lie algebra of the poincare group and they lead to the correct transformation properties for the canonical variables we stress the importance of a canonical realization of the full symmetry group of the evolution equations the covariance of the theory under the symmetry group can be used as a criterion to discriminate among different canonical structures for the evolution equations

a dynamical adaptive tensor method for the vlasov poisson system abstract a numerical method is proposed to solve the full eulerian time dependent vlasov poisson system the algorithm relies on the construction of a tensor decomposition of the solution whose rank is adapted at each time step this decomposition is obtained through the use of an efficient modified progressive generalized decomposition pgd method whose convergence is proved we suggest in addition a symplectic time discretization splitting scheme that preserves the hamiltonian properties of the system this scheme is naturally obtained by considering the tensor structure of the approximation the proposed approach is illustrated through time dependent 1d 1d 2d 2d and 3d 3d numerical examples

symplectic and multisymplectic numerical methods for maxwell s equations in this paper we compare the behaviour of one symplectic and three multisymplectic methods for maxwell s equations in a simple medium this is a system of pdes with symplectic and multisymplectic structures we give a theoretical discussion of how some numerical methods preserve the discrete versions of the local and global conservation laws and verify this behaviour in numerical experiments we also show that these numerical methods preserve the divergence furthermore we extend the discussion on dispersion for multi symplectic methods applied to pdes with one spatial dimension to include anisotropy when applying multi symplectic methods to maxwell s equations in two spatial dimensions lastly we demonstrate how varying the courant friedrichs lewy cfl number can cause the multi symplectic methods in our comparison to behave differently which can be explained by the study of backward error analysis for pdes

symmetry and bifurcations of momentum mappings the zero set of a momentum mapping is shown to have a singularity at each point with symmetry the zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function the proof uses the kuranishi theory of deformations among the applications it is shown that the set of all solutions of the yang mills equations on a lorentz manifold has a singularity at any solution with symmetry in the sense of a pure gauge symmetry similarly the set of solutions of einstein s equations has a singularity at any solution that has spacelike killing fields provided the spacetime has a compact cauchy surface

splitting k symplectic methods for non canonical separable hamiltonian problems non canonical hamiltonian systems have k symplectic structures which are preserved by k symplectic numerical integrators there is no universal method to construct k symplectic integrators for arbitrary non canonical hamiltonian systems however in many cases of interest by using splitting we can construct explicit k symplectic methods for separable non canonical systems in this paper we identify situations where splitting k symplectic methods can be constructed comparative numerical experiments in three non canonical hamiltonian problems show that symmetric non symmetric splitting k symplectic methods applied to the non canonical systems are more efficient than the same order gauss methods non symmetric symplectic methods applied to the corresponding canonicalized systems for the non canonical lotka volterra model the splitting algorithms behave better in efficiency and energy conservation than the k symplectic method we construct via generating function technique in our numerical experiments the favorable energy conservation property of the splitting k symplectic methods is apparent

coadjoint orbits vortices and clebsch variables for incompressible fluids this paper is a study of incompressible fluids especially their clebsch variables and vortices using symplectic geometry and the lie poisson structure on the dual of a lie algebra following ideas of arnold and others it is shown that euler s equations are lie poisson equations associated to the group of volume preserving diffeomorphisms the dual of the lie algebra is seen to be the space of vortices and kelvin s circulation theorem is interpreted as preservation of coadjoint orbits in this context clebsch variables can be understood as momentum maps the motion of n point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit and the standard canonical variables for them are a special kind of clebsch variables point vortices with cores vortex patches and vortex filaments can be understood in a similar way this leads to an explanation of the geometry behind the hald beale majda convergence theorems for vorticity algorithms symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the kdv and the schrodinger equations respectively

a rigorous derivation of the hamiltonian structure for the nonlinear schrodinger equation abstract we consider the cubic nonlinear schrodinger equation nls in any spatial dimension which is a well known example of an infinite dimensional hamiltonian system inspired by the knowledge that the nls is an effective equation for a system of interacting bosons as the particle number tends to infinity we provide a derivation of the hamiltonian structure which is comprised of both a hamiltonian functional and a weak symplectic structure for the nonlinear schrodinger equation from quantum many body systems our geometric constructions are based on a quantized version of the poisson structure introduced by marsden morrison and weinstein 24 for a system describing the evolution of finitely many indistinguishable classical particles

prediction of evanescent coupling efficiency in two parallel silica nanowires abstract in this study the efficiency of evanescent wave coupling between two air clad silica nanowires in single mode operation is numerically predicted in time domain using the finite difference method to this end a three dimensional scheme developed for solving the maxwell s equations on staggered grids will be employed the electric and magnetic field solutions are sought subjected to the zero divergence condition or gauss s law in discrete level in addition we aim to conserve hamiltonians existing in ideal maxwell s equations at all times by adopting the explicit second order accurate symplectic partitioned runge kutta temporal scheme moreover all the spatial derivative terms in the faraday s and ampere s equations are approximated by the proposed scheme which can render not only fourth order spatial accuracy but also minimize the discrepancy between the exact and the derived numerical phase velocities

new multisymplectic self adjoint scheme and its composition scheme for the time domain maxwell s equations in this paper we investigate euler box scheme for bridges multisymplectic form of maxwell s equations a new multisymplectic scheme is derived for maxwell s equations we prove that it is also a self adjoint scheme in time direction the multisymplecticity of composition schemes based on the new scheme is also discussed two numerical examples are proposed to indicate that the derived multisymplectic schemes are effective when used to integrate the 2 1 dimensional maxwell s equations

casimir invariants and the jacobi identity in dirac s theory of constrained hamiltonian systems we consider constrained hamiltonian systems in the framework of dirac s theory we show that the jacobi identity results from imposing that the constraints are casimir invariants regardless of whether the matrix of poisson brackets between constraints is invertible or not we point out that the proof we provide ensures the validity of the jacobi identity everywhere in the phase space and not just on the surface defined by the constraints two examples are considered a finite dimensional system with an odd number of constraints and the vlasov poisson reduction from vlasov maxwell equations

hamiltonian mechanics of complex fluids constitutive relations for complex fluids e g polymeric fluids are derived from the assumption that the equations that govern the time evolution of the complex fluids possess the generalised hamiltonian structure the constitutive relations consist of a configuration space kinetic equation an extended local equilibrium relation an expression for the extra stress tensor and an expression for the heat flux in the particularly interesting case of complex fluids with spatially inhomogeneous internal structure the extra stress tensor includes an extension of the eriksen stress and the heat flux includes non fourier type contributions

the hamiltonian structure of the maxwell vlasov equations morrison 25 has observed that the maxwell vlasov and poisson vlasov equations for a collisionless plasma can be written in hamiltonian form relative to a certain poisson bracket we derive another poisson structure for these equations by using general methods of symplectic geometry the main ingredients in our construction are the symplectic structure on the co adjoint orbits for the group of canonical transformations and the symplectic structure for the phase space of the electromagnetic field regarded as a gauge theory our poisson bracket satisfies the jacobi identity whereas morrison s does not 37 our construction also shows where canonical variables can be found and can be applied to the yang mills vlasov equations and to electromagnetic fluid dynamics

the hamiltonian structure of classical chromohydrodynamics noncanonical hamiltonian structures are presented both for yang mills vlasov plasmas and for ideal fluids interacting with yang mills fields the hamiltonian structure for the yang mills vlasov system passes over to that for the yang mills fluid in the cold plasma limit the resulting hamiltonian structure is shown to correspond to a lie algebra

noncanonical poisson brackets for elastic and micromorphic solids this paper investigates the lagrangian to eulerian transformation approach to the construction of noncanonical poisson brackets for the conservative part of elastic solids and micromorphic elastic solids the dirac delta function links lagrangian canonical variables and eulerian state variables producing noncanonical poisson brackets from the corresponding canonical brackets specifying the hamiltonian functionals generates the evolution equations for these state variables from the poisson brackets different elastic strain tensors such as the green deformation tensor the cauchy deformation tensor and the higher order deformation tensor are appropriate state variables in poisson bracket formalism since they are quantities composed of the deformation gradient this paper also considers deformable directors to comprise the three elastic strain density measures for micromorphic solids furthermore the technique of variable transformation is also discussed when a state variable is not conserved along with the motion of the body

the differentiable structure of three remarkable diffeomorphism groups the goal of this paper is to find the lie groups for three well known lie algebras the globally hamiltonian vector fields the infinitesimal quantomorphisms and the homogeneous real valued functions of degree one on the cotangent bundle minus the zero section in all our considerations the underlying manifold is assumed to be compact without boundary before explaining and motivating our results a very brief review of the lie group structure of diffeomorphism groups is in order 9 10 18 if m n are compact boundaryless finite dimensional manifolds rn tn n the tangent bundle of n an hs map s dimm 2 from m to n has by definition all derivatives of order dim m 2 then the space ip m n of all such maps is a hilbert manifold whose tangent space at every point is the hilbert space t i h m n gelid m tn i n g f if e m is a vector bundle the same construction works for all hs sections hs e of e s dim m 2 if n ir hs m n will be denoted by c m ir let i m denote the diffeomorphisms of m of sobolev class h s l i e r t l m if and only if t is bijective and t t l m m are of class h 1 s l m is a topological group and since it is open in hs i m m it is also a hilbert manifold right multiplication r m i m r o t is c for each t y m and if t e s k l m left multiplication l t m i m l t o is c k the inversion map t t 1 in t m is only continuous the tangent maps of r and l at e the identity of a m are given by tr tel x t ox where x yys i m hs i tm the set of all h l vector fields on m the tangent space at e tr coincides with y m which is a sobolev space the bracket x y of x y f s i m is however only of class h one derivative is lost the usual bracket of vector fields is the lie algebra bracket of 5f x m i e if j

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

numerical analysis of a multi symplectic scheme for the time domain maxwell s equations in this paper we analyze qualitative properties of the first multi symplectic scheme for two dimensional maxwell s equations we prove that the scheme is unconditionally stable and convergent non dissipative and divergence free the numerical dispersion relation of the scheme is shown to converge to the exact dispersion relation of the maxwell equations we also present some numerical results to confirm our theoretical results

the maxwell vlasov equations in euler poincare form low s well known action principle for the maxwell vlasov equations of ideal plasma dynamics was originally expressed in terms of a mixture of eulerian and lagrangian variables by imposing suitable constraints on the variations and analyzing invariance properties of the lagrangian as one does for the euler equations for the rigid body and ideal fluids we first transform this action principle into purely eulerian variables hamilton s principle for the eulerian description of low s action principle then casts the maxwell vlasov equations into euler poincare form for right invariant motion on the diffeomorphism group of position velocity phase space openface r 6 legendre transforming the eulerian form of low s action principle produces the hamiltonian formulation of these equations in the eulerian description since it arises from euler poincare equations this hamiltonian formulation can be written in terms of a poisson structure that contains the lie poisson bracket on the dual of a semidirect product lie algebra because of degeneracies in the lagrangian the legendre transform is dealt with using the dirac theory of constraints another maxwell vlasov poisson structure is known whose ingredients are the lie poisson bracket on the dual of the lie algebra of symplectomorphisms of phase space and the born infeld brackets for the maxwell field we discuss the relationship between these two hamiltonian formulations we also discuss the general kelvin noether theorem for euler poincare equations and its meaning in the plasma context

the diameter of the symplectomorphism group is infinite shnirelman has proved that the group of volume preversing diffeomorphisms of the cube in r 3 has finite diameter and has announced the result that this is false for the square despite the fact that shnirelman formulated his theorem only for the 3 dimensional cube his proof can be modified for the case of the group of volume preserving diffeomorphisms of any compact simply connected riemannian manifold of dimension 2 it turns out that the situation with the group of symplectic diffeomorphisms is completely different we prove that the diameter of the symplectomorphism group of any compact exact symplectic manifold necessarily with boundary is infinite

hamiltonian formalism for general relativistic adiabatic fluids abstract we derive the hamiltonian structures of three theories non relativistic special relativistic and general relativistic adiabatic fluids each in the eulerian representation in riemannian space or lorentzian spacetime all by the same procedure using standard variational principles the evolution in each case is generated by a hamiltonian that is equivalent to that obtained from a canonical analysis for the gravitational variables the poisson bracket has the usual canonical symplectic structure however for the fluid variables the three theories all share the same lie poisson bracket when expressed in the appropriate spaces of physical variables constructed here this shared lie poisson bracket is associated to the dual of the semidirect product lie algebra of vector fields acting on differential forms an immediate consequence of this shared structure is that each of these theories possesses an infinite family of conservation laws the so called casimirs that belong to the kernel of the lie poisson bracket the role of these casimirs in the study of lyapunov stability or dynamic stability for fluid equilibria is discussed the relationship of this approach to other approaches in the literature is also discussed

the hamiltonian structure of nonlinear elasticity the material and convective representations of solids rods and plates it is our belief that a thorough understanding of the mathematical underpinnings of elasticity is crucial to its analytical and numerical implementation for example in the analysis of rotating structures the coupling of the equations for geometrically inexact models obtained by linearization or other approximations with those for rotating rigid bodies can easily lead to misleading artificial softening effects that can significantly alter numerical results see simo and vuquoc 1986c especially equations 3 and 5 in this paper we consider fully nonlinear geometrically exact models for rods plates and shells which take into account shear and torsion as well as the usual bending effects in traditional rod and plate models these models can be obtained either from the three dimensional theory by a systematic use of projection methods see e g antman 1972 and naghdi 1972 or by a direct approach within the context of cosserat continuum remarkably the two approaches lead to essentially the same form of the governing field equations in the present context we have chosen as a model problem a particular rod model which may be regarded as an extension of the classical kirchhoff love model see love 1944 to include shear deformations as in

hamiltonian splitting for the vlasov maxwell equations a new splitting is proposed for solving the vlasov maxwell system this splitting is based on a decomposition of the hamiltonian of the vlasov maxwell system and allows for the construction of arbitrary high order methods by composition independent of the specific deterministic method used for the discretization of the phase space moreover we show that for a spectral method in space this scheme satisfies poisson s equation without explicitly solving it finally we present some examples in the context of the time evolution of an electromagnetic plasma instability which emphasizes the excellent behavior of the new splitting compared to methods from the literature

some geometric calculations on wasserstein space we compute the riemannian connection and curvature for the wasserstein space of a smooth compact riemannian manifold

multisymplectic preissman scheme for the time domain maxwell s equations from the bridges multisymplectic form of maxwell s equations we derive a multisymplectic preissman scheme which couples two time levels for 2 1 dimensional maxwell s equations the scheme is proven to preserve the discrete local energy exactly numerical results are reported to illustrate that the scheme is effective and it can get more precise numerical solutions than yee s scheme our numerical results can also indicate that the scheme keeps the discrete local energy and the global energy very well

automorphism groups on normal singular cubic surfaces with no parameters the classification of normal singular cubic surfaces in p 3 over a complex number field c was given by j w bruce and c t c wall in this paper first we prove their results by a different way second we provide normal forms of normal singular cubic surfaces according to the type of singularities and finally we determine automorphism groups on normal singular cubic surfaces with no parameters

some twisted self dual solutions for the yang mills equations on a hypertorus thesu n yang mills equations are considered in a four dimensional euclidean box with periodic boundary conditions hypertorus gauge invariant twists can be introduced in these boundary conditions to be labeled with integersn n defined modulon the pontryagin number in this space is often fractional whenever this number is zero there are solutions to the equationsg 0 hereg is the covariant curl when this number is not zero we find a set of solutions to the equations g mu nu tilde g mu nu provided that the periodsa of the box satisfy certain relations

classification and casimir invariants of lie poisson brackets abstract we classify lie poisson brackets that are formed from lie algebra extensions the problem is relevant because many physical systems owe their hamiltonian structure to such brackets a classification involves reducing all brackets to a set of normal forms and is achieved partially through the use of lie algebra cohomology for extensions of order less than five the number of normal forms is small and they involve no free parameters we derive a general method of finding casimir invariants of lie poisson bracket extensions the casimir invariants of all low order brackets are explicitly computed we treat in detail a four field model of compressible reduced magnetohydrodynamics

development of a 3d staggered fdtd scheme for solving maxwell s equations in drude medium an explicit finite difference scheme is developed to solve the three dimensional maxwell s equations in drude medium our aim of developing this scheme in time domain is to compute solutions in staggered grids from the faraday s and ampere s equations the electric and magnetic field solutions are sought subject to the discrete zero divergence condition gauss s law the local conservation laws in ideal maxwell s equations are also numerically preserved all the time using the explicit second order accurate symplectic partitioned runge kutta temporal scheme the spatial derivative terms in the faraday s and ampere s equations are discretized to obtain fourth order accuracy using the developed scheme underlying the concept of minimizing the discrepancy between the exact and the derived numerical phase velocities dispersion and anisotropy errors have been much reduced through the procedure of minimizing phase velocity error in addition to performing the fundamental analysis on the proposed scheme the computational efficiency and long term accurate properties embedded in the proposed symplectic dispersion error reduction centered scheme are numerically demonstrated through several test problems investigated in ideal and drude media

geometric variational discretization of continuum theories this study derives geometric variational discretization of continuum theories arising in fluid dynamics magnetohydrodynamics mhd and the dynamics of complex fluids a central role in these discretizations is played by the geometric formulation of fluid dynamics which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume preserving diffeomorphisms of the fluid domain inspired by this framework we construct a finite dimensional approximation to the diffeomorphism group and its lie algebra thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group the extension to mhd and complex fluid flow is then made through an appeal to the theory of euler poincare systems with advection which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters upon deriving a family of structured integrators for these systems we test their performance via a numerical implementation of the update schemes on a cartesian grid among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries automatic satisfaction of solenoidal constraints on vector fields good long term energy behavior robustness with respect to the spatial and temporal resolution of the discretization and applicability to irregular meshes

cylindrically invariant flows in relativistic fluid dynamics abstract we show that partially cylindrically symmetric solutions of the motion equations of special relativistic adiabatic ideal fluid dynamics have the same noncanonical hamiltonian structure as the original motion equations

hyperkahler structures and infinite dimensional grassmannians abstract in this paper we describe an example of a hyperkahler quotient of a banach manifold by a banach lie group although the initial manifold is not diffeomorphic to a hilbert manifold not even to a manifold modelled on a reflexive banach space the quotient space obtained is a hilbert manifold which can be furthermore identified either with the cotangent space of a connected component gr res j j z of the restricted grassmannian or with a natural complexification of this connected component thus proving that these two manifolds are isomorphic hyperkahler manifolds moreover kahler potentials associated with the natural complex structure of the cotangent space of gr res j and with the natural complex structure of the complexification of gr res j are computed using kostant souriau s theory of prequantization

reduction theory and the lagrange routh equations reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of euler jacobi lagrange hamilton routh poincare and others the modern vision of mechanics includes besides the traditional mechanics of particles and rigid bodies field theories such as electromagnetism fluid mechanics plasma physics solid mechanics as well as quantum mechanics and relativistic theories including gravity symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas to subtle symmetries underlying integrable systems reduction theory concerns the removal of symmetries and their associated conservation laws variational principles along with symplectic and poisson geometry provide fundamental tools for this endeavor reduction theory has been extremely useful in a wide variety of areas from a deeper understanding of many physical theories including new variational and poisson structures to stability theory integrable systems as well as geometric phases this paper surveys progress in selected topics in reduction theory especially those of the last few decades as well as presenting new results on non abelian routh reduction we develop the geometry of the associated lagrange routh equations in some detail the paper puts the new results in the general context of reduction theory and discusses some future directions

a partial differential equation with infinitely many periodic orbits chaotic oscillations of a forced beam this paper delineates a class of time periodically perturved evolution equations in a banach space whose associated poincar e map contains a smale horseshoe this implies that such systems possess periodic orbits with arbitrarily high period the method uses techniques originally due to melnikov and applies to systems of the form x f0 x f1 x t where x f0 x is hamiltonian and has a homoclinic orbit we give an example from structural mechanics sinusoidally forced vibrations of a buckled beam

non canonical poisson bracket for nonlinear elasticity with extensions to viscoelasticity the authors introduce a generalized bracket which is capable of generating the dynamical equations governing the flow of both elastic and viscoelastic media this generalized bracket is divided into two parts a noncanonical poisson bracket and a new dissipation bracket the non canonical poisson bracket is the eulerian equivalent of the canonical lagrangian poisson bracket corresponding to an ideal non dissipative continuum it is derived for a nonlinear elastic medium and then it is used to obtain the eulerian equations of motion in nonlinear elasticity valid for large deformations it is shown that the proposed non canonical bracket naturally leads to a materially objective relation involving the upper convected time derivative of a strain tensor as suggested by oldroyd 40 years ago the dissipation bracket for linear irreversible thermodynamics is next proposed in a general phenomenological circumstance so that the dissipation processes occurring in real systems viscous and relaxation phenomena can be incorporated into the hamiltonian formalism this bracket diverges from previously proposed dissipation brackets in that it uses the same generating functional i e the hamiltonian as the poisson bracket rather than an entropy functional or a dissipative potential it is shown that in combination with the choice of an appropriate hamiltonian functional the generalized bracket proposed here can generate the governing equations for many viscoelastic media including the voigt solid and the maxwell viscoelastic fluid

higher order symmetries of the compressible one dimensional isentropic fluid equations the equations of compressible one dimensional isentropic flow are shown to have infinitely many higher order symmetries by means of recursion operators furthermore a new first order conserved functional is found to be associated with a third order symmetry via noether s theorem

new solutions of the jacobi equations for three dimensional poisson structures a systematic investigation of the skew symmetric solutions of the three dimensional jacobi equations is presented as a result three disjoint and complementary new families of solutions are characterized such families are very general thus unifying many different and well known poisson structures seemingly unrelated which now appear embraced as particular cases of a more general solution this unification is not only conceptual but allows the development of algorithms for the explicit determination of important properties such as the symplectic structure the casimir invariants and the darboux canonical form which are known only for a limited sample of poisson structures these common procedures are thus simultaneously valid for all the particular cases which can now be analyzed in a unified and more economic framework instead of using a case by case approach in addition the methods developed are valid globally in phase space thus ameliorating the usual scope of darboux reduction which is only of

the hamiltonian structure of the modified and fifth order korteweg de vries equation dirac s theory of constraints is used to derive the hamiltonian structure of the modified and fifth order korteweg de vries equations a transformation to physical variables is performed on the canonical structure and is shown to be equivalent to the dirac bracket

exponential integrators for stochastic maxwell s equations driven by ito noise abstract this article presents explicit exponential integrators for stochastic maxwell s equations driven by both multiplicative and additive noises by utilizing the regularity estimate of the mild solution we first prove that the strong order of the numerical approximation is 1 2 for general multiplicative noise combining a proper decomposition with the stochastic fubini s theorem the strong order of the proposed scheme is shown to be 1 for additive noise moreover for linear stochastic maxwell s equation with additive noise the proposed time integrator is shown to preserve exactly the symplectic structure the evolution of the energy as well as the evolution of the divergence in the sense of expectation several numerical experiments are presented in order to verify our theoretical findings

statistical mechanics with non integrable topological constraints self organization in knotted phase space the object of this study is the statistical mechanics of dynamical systems lacking a hamiltonian structure due to the presence of non integrable topological constraints that limit the accessible regions of the phase space focusing on the simplest three dimensional case we develop a procedure poissonization that assigns to any three dimensional non hamiltonian system an equivalent four dimensional hamiltonian system endowed with a proper time the statistical distribution is then constructed in the recovered four dimensional canonical phase space projecting in the original reference frame we show that the statistical distribution departs from standard maxwell boltzmann statistics the deviation is a function of the knottedness of the phase space which is measured by the helicity density of the topological constraint the theory is then generalized to a class of non hamiltonian systems in higher dimensions

lin constraints clebsch potentials and variational principles the poisson bracket formulation of fluid plasma and rigid body type systems has undergone considerable recent development using techniques of symmetry group reduction the relationship between this approach and that using lin constraints and clebsch potentials is established the connection is made in the setting of abstract clebsch variables as well as that of variational principles on reduced spaces variational principles for both the clebsch and reduced form such as fluids in spatial representation are derived from the standard variational principle of hamilton in material lagrangian representation using reduction theory

numerical dispersion analysis of a multi symplectic scheme for the three dimensional maxwell s equations in this paper we study a multi symplectic scheme for three dimensional maxwell s equations in a simple medium this is a system of pdes with multi symplectic structures we prove that this multi symplectic scheme preserves the discrete version of local and global energy conservation law and the discrete divergence furthermore we extend the discussion to several dispersion properties of the multi symplectic scheme including the numerical dispersion relation the numerical group velocity the effect of large time steps and the cfl condition

poisson brackets and clebsch representations for magnetohydrodynamics multifluid plasmas and elasticity abstract poisson brackets are constructed by the same mathematical procedure for three physical theories ideal magnetohydrodynamics multifluid plasmas and elasticity each of these brackets is given a simple lie algebraic interpretation moreover each bracket is induced to physical space by use of a canonical poisson bracket in the space of clebsch potentials which are constructed for each physical theory by the standard procedure of constrained lagrangians

extended kinetic theory the domain of applicability of one point kinetic theory is enlarged by admitting two point distribution functions or some of their moments as extra state variables kinetic equations governing the time evolution in the extended state space are introduced as particular realizations of the nonlinear onsager casimir equation this abstract equation collects the common structure of equations arising in the analysis of compatibility of two levels of description e g kinetic theory and equilibrium thermodynamics or kinetic theory and hydrodynamics the structure consists of a poisson bracket and two potentials the poisson bracket expresses kinematics the first potential called the thermodynamic potential generates the time evolution and links the theory with thermodynamics the second potential called the dissipative potential introduces dissipation into the time evolution that is compatible with the kinematics and thermodynamics to find a particular realization of the nonlinear onsager casimir equati

hamiltonian dynamics and stability analysis of neutral electromagnetic fluids with induction abstract the lie poisson hamiltonian structure of the special relativistic electromagnetic fluid equations is derived this hamiltonian structure provides synthesis and insight leading to new conservation laws and stability conditions for the equilibrium solutions a corollary of the stability results generalizes rayleigh s inflectional instability criterion for ideal incompressible fluids to the present case another alternative hamiltonian formulation of relativistic electromagnetic fluid dynamics is constructed systematically via lie algebraic considerations of the poisson bracket in particular relativistic magnetohydrodynamics emerges naturally from these considerations the nonrelativistic limits of these two formulations are also determined and are shown to be regular and to preserve the corresponding lie poisson structures

form invariant poisson brackets of hydrodynamic type with several spatial variables form invariant poisson brackets of hydrodynamic type with several spatial variables x1 xp are introduced for any commuting vector fields u1 um on a manifold mn the incompressibility equation for certain parameters of the poisson brackets is derived poisson brackets connected with an arbitrary dynamical system on mn are constructed a gauge transform for the poisson brackets with one spatial variable x is defined and its action onto the tensor invariants is studied a general identity between the coefficients bkij u and gij u is found