time discretization of hamiltonian dynamical systems abstract difference equations for hamiltonian systems are derived from a discrete variational principle the difference equations completely determine piecewise linear continuous trajectories which exactly conserve the hamiltonian function at the midpoints of each linear segment a generating function exists for transformations between the vertices of the trajectories existence and uniqueness results are present as well as simulation results for a simple pendulum and an inverse square law system

the complex geometry of weak piecewise smooth solutions of integrable nonlinear pde s of shallow water and dym type an extension of the algebraic geometric method for nonlinear integrable pde s is shown to lead to new piecewise smooth weak solutions of a class of n component systems of nonlinear evolution equations this class includes among others equations from the dym and shallow water equation hierarchies the main goal of the paper is to give explicit theta functional expressions for piecewise smooth weak solutions of these nonlinear pde s which are associated to nonlinear subvarieties of hyperelliptic jacobians

complete planar impedance conservation and pml boundary value solutions this paper shows that a straightforward way to evaluate planar impedance conservation boundary values of arbitrary boundary operators and orientations is to directly copy the construction from the well understood 1 d robin impedance matching solution and then let the boundary rotate freely copying from this classical solution hence requires no use of split field and any of its stretched coordinate reformulations in fact it actually gives the complete general superposition solution which also explains that the existing pml considers only a rational boundary operator and solves only a coordinately aligned example the construction thus embraces both general order polynomial and nonpolynomial boundary operators and noncoordinately aligned boundary solutions to be put on top of the pml orthogonal terminations a constructive source field energy coupling relation is also introduced to simplify the impedance conservation analysis to read as the triple product identity evident by the various forms of maxwell equations we elaborate in detail that boundary value equations can also be written in various equivalent and endless redundant forms

geometric description of vakonomic and nonholonomic dynamics comparison of solutions we treat the vakonomic dynamics with general constraints within a new geometric framework which can be useful in the study of optimal control problems we compare our formulation with the one of vershik and gershkovich in the case of linear constraints we show how nonholonomic mechanics also admits a new geometrical description which allows us to develop an algorithm of comparison between the solutions of both dynamics examples illustrating the theory are treated

on variations in discrete mechanics and field theory in this paper we propose the difference variational approach with variable step length for discrete mechanics and field theory based upon the hamilton s principle for the vertical variations and the horizontal variations we get the equation of motion and the relevant relations between the equation of motion and conservation law of the energy energy momentum tensor for continuous as well as for the discrete systems respectively for those discrete cases these relations determine the variable step length of the differences in addition by taking the double operation of vertical exterior differential on action we show that there should exist the euler lagrange cohomology for both continuous and variable step length difference cases in general this indicates that the necessary and sufficient condition for symplectic multisymplectic structure preserving properties is the relevant euler lagrange one form being closed in each case our approach can be applied to both the lagrangian formalism and the hamiltonian formalism via discrete legendre transformation for the difference discrete mechanics and field theory c 2003 american institute of physics

a variationally consistent fractional time step integration method for incompressible and nearly incompressible lagrangian dynamics in this paper we present a fractional time step method for lagrangian formulations of solid dynamics problems the method can be interpreted as belonging to the class of variational integrators which are designed to conserve linear and angular momentum of the entire mechanical system exactly energy fluctuations are found to be minimal and stay bounded for long durations in order to handle incompressibility a mixed formulation in which the pressure appears explicitly is adopted the velocity update over a time step is split into deviatoric and volumetric components the deviatoric component is advanced using explicit time marching whereas the pressure correction for each time step is computed implicitly by solving a poisson like equation once the pressure is known the volumetric component of the velocity update is calculated in contrast with standard explicit schemes where the time step size is determined by the speed of the pressure waves the allowable time step for the proposed scheme is found to depend only on the shear wave speed this leads to a significant advantage in the case of nearly incompressible materials and permits the solution of truly incompressible problems copyright 2005 john wiley sons ltd

fdtd discrete planewave fdtd dpw formulation for a perfectly matched source in tfsf simulations a technique is proposed for the generation of planewaves in the total field scattered field tfsf formulation of the fdtd method the method is developed using the 1d properties of a planewave and optimized projection of the 3d finite difference operators to the 1d domain the result is an efficient and accurate planewave source that can be propagated on six 1d grids concurrent with the main simulation and that is perfectly matched to the main 2d 3d fdtd domain for any source function numerical simulations show that the technique is valid for any angle of propagation and for any gridcell aspect ratio with non physical reflections in the scattered field domain on the order of machine precision 300 db

objective energy momentum conserving integration for the constrained dynamics of geometrically exact beams in this paper the results in s leyendecker p betsch p steinmann energy conserving integration of constrained hamiltonian systems a comparison of approaches comput mech 33 2004 174 185 are extended to geometrically exact beams the finite element formulation for nonlinear beams in terms of directors providing a framework for the objective description of their dynamics is considered geometrically exact beams are analysed as hamiltonian systems subject to holonomic constraints with a hamiltonian being invariant under the action of so 3 the reparametrisation of the hamiltonian in terms of the invariants of so 3 is perfectly suited for a temporal discretisation which leads to energy momentum conserving integration in this connection the influence of alternative procedures the lagrange multiplier method the penalty method and the augmented lagrange method for the treatment of the constraints is investigated for the example of a beam with concentrated masses 2005 elsevier b v all rights reserved

local space time dirichlet neumann nonreflecting boundary values half space spatial orthogonal mode projections this paper follows the time honored analyses to provide a formal bridge connecting the pml and the well understood classical dirichlet and neumann boundary value problems the connection offers many theoretical and practical advantages not immediately obvious from any pml reconstructions for example the analysis shows that the reflection free of pml boundaries is composed of the superposition of one dirichlet and one neumann solution acting out of phase constituents in fact it uncovers that the existing pml solutions are readily obtained from the projection of the well understood 1 d robin impedance matched wave equations the projection hence not only explains how the pml split fields fundamentally come about but also that the pml implicitly uses only an orthogonal projection naturally we further extend the construction to include nonorthogonal decomposition local to global coordinate transformation is introduced to give a general nonreflecting boundary value solution valid for planar orientations not aligned with the cartesian coordinates nonorthogonal generalizations also make it geometrically explicit as to what causes an incorrect pml problem that can arise from mediums lacking orthogonal symmetry invariant terminations

simultaneous space time adaptive wavelet solution of nonlinear parabolic differential equations dynamically adaptive numerical methods have been developed to efficiently solve differential equations whose solutions are intermittent in both space and time these methods combine an adjustable time step with a spatial grid that adapts to spatial intermittency at a fixed time the same time step is used for all spatial locations and all scales this approach clearly does not fully exploit space time intermittency we propose an adaptive wavelet collocation method for solving highly intermittent problems e g turbulence on a simultaneous space time computational domain which naturally adapts both the space and time resolution to match the solution besides generating a near optimal grid for the full space time solution this approach also allows the global time integration error to be controlled the efficiency and accuracy of the method is demonstrated by applying it to several highly intermittent 1d t dimensional and 2d t dimensional test problems in particular we found that the space time method uses roughly 18 times fewer space time grid points and is roughly 4 times faster than a dynamically adaptive explicit time marching method while achieving similar global accuracy

connecting pml to classical robin boundary value problems in this paper we show that pml solutions are most readily derived from the well understood 1 d robin boundary value classically defined as a weighted combination of dirichlet and neumann constituents the derivation thus uses no split field or stretched coordinate reformulation continuing with this classical construction in fact gives the complete superposition solution valid for arbitrary boundary orientations and operators this further enables the explanation that uniaxial pml matrix solves only the coordinately aligned terminations temporal implicit and spatial explicit representations are presented solving robin impedance matched solutions thus amounts to merely relabeling the corresponding boundary variable name following the superposition principle we demonstrate mathematically geometrically and numerically that the pml reflection free boundary is also composed of one dirichlet and one neumann acting out of phase solutions relating this to reflectional symmetry is presented

intertwining operators associated to the group for any finite reflection group g on an euclidean space there is a parametrized commutative algebra of differential difference operators with as many parameters as there are conjugacy classes of reflections in g there exists a linear isomorphism on polynomials which intertwines this algebra with the algebra of partial differential operators with constant coefficients for all but a singular set of parameter values containing only certain negative rational numbers this paper constructs an integral transform implementing the intertwining operator for the group s3 the symmetric group on three objects for parameter value the transform is realized as an absolutely continuous measure on a compact subset of m2 r which contains the group as a subset of its boundary the construction of the integral formula involves integration over the unitary group u 3 associated to any finite reflection group g on an euclidean space there is a parametrized commutative algebra of differential difference operators with as many parameters as there are conjugacy classes of reflections in g it has been shown that there exists a linear isomorphism on polynomials which intertwines this algebra with the algebra of partial differential operators with constant coefficients for all but a singular set of parameter values this singular set contains only negative values and is closely linked to the zero set of the poincare series of g this paper constructs an integral transform implementing the intertwining map for the group s3 the symmetric group on three objects for positive parameter values previously this had been done only for the group z2 acting by sign change on r where the transform is a classical fractional integral the transform in this paper has its origin in the adjoint action of the unitary group u 3 on the linear space of real diagonal 3 x 3 matrices the complexification of the maximal torus this will lead to a transform realized as an absolutely continuous measure on a certain compact subset of m2 r here is a concise statement of the main result rephrased from formulas 5 1 5 6 the operator v intertwines the differential difference operators received by the editors august 22 1994 1991 mathematics subject classification primary 22e30 33c80 secondary 33c50 20b30

symplectic integration of constrained hamiltonian systems by composition methods recent work reported in the literature suggests that for the long term integration of hamiltonian dynamical systems one should use methods that preserve the symplectic structure of the flow in this paper the symplecticity of numerical integrators is investigated for constrained hamiltonian systems with holonomic constraints the following two results will be derived i it is shown that any first or second order symplectic integrator for unconstrained problems can be generalized to constrained systems such that the resulting scheme is symplectic and preserves the constraints based on this higher order methods can be derived by the same composition methods used for unconstrained problems ii leimkuhler and reich math comp 63 1994 pp 589 605 derived symplectic integrators based on dirac s reformulation of the constrained problem as an unconstrained hamiltonian system however although the unconstrained reformulation can be handled by direct application of any symplectic implicit runge kutta m

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

symplectic energy momentum preserving variational integrators the purpose of this paper is to develop variational integrators for conservative mechanical systems that are symplectic and energy and momentum conserving to do this a space time view of variational integrators is employed and time step adaptation is used to impose the constraint of conservation of energy criteria for the solvability of the time steps and some numerical examples are given

paley wiener theorems for the dunkl transform we conjecture a geometrical form of the paley wiener theorem for the dunkl transform and prove three instances thereof by using a reduction to the one dimensional even case shift operators and a limit transition from opdam s results for the graded hecke algebra respectively these paley wiener theorems are used to extend dunkl s intertwining operator to arbitrary smooth functions furthermore the connection between dunkl operators and the cartan mo tion group is established it is shown how the algebra of radial parts of in variant differential operators can be described explicitly in terms of dunkl operators this description implies that the generalized bessel functions coin cide with the spherical functions in this context of the cartan motion group the restriction of dunkl s intertwining operator to the invariants can be inter preted in terms of the abel transform we also show that for certain values of the multiplicities of the restricted roots the abel transform is essentially inverted by a differential operator 1 introduction and overview in recent times the study of special functions associated with root systems has developed to a considerable degree starting with a number of conjectures by mac donald and the work of heckman and opdam on multivariable hypergeometric functions in the late 1980 s the development of the theory was greatly enhanced by the introduction of rational dunkl operators by dunkl 6 through various intermediate steps of generalization these operators can even be said to have ulti mately provided crucial building blocks for cherednik s work on double affine hecke algebras and the q macdonald conjectures originally before the introduction of dunkl operators the idea when studying special functions related to root systems was to consider root multiplicities in the theory of spherical functions on lie groups as parameters and then to develop a theory for weyl group invariant objects in this more general situation without the aid of the presence of the group it was this point of view which underlay the macdonald conjectures and which led heckman and opdam to the development of their theory of hypergeometric functions in higher dimension one of the main technical problems in this context is the description of the generalized radial parts of invariant differential operators apart from an explicit formula for the generalized radial part of the laplacian an expression which was in fact the starting point for heckman and opdam the other operators remain somewhat intangible

symplectic partitioned runge kutta methods for constrained hamiltonian systems this article deals with the numerical treatment of hamiltonian systems with holonomic constraints a class of partitioned runge kutta methods consisting of the couples of s stage lobatto iiia and lobatto iiib methods has been discovered to solve these problems efficiently these methods are symplectic preserve all un derlying constraints and are superconvergent with order 2s 2 for separable hamiltonians of the form h q p frac 1 2 p t m 1 p u q the rattle algorithm based on the verlet method was up to now the only known symplectic method preserving the constraints in fact this method turns out to be equivalent to the 2 stage lobatto iiia iiib method of order 2 numerical examples have been performed which illustrate the theoretical results

long term dissipativity of time stepping algorithms for an abstract evolution equation with applications to the incompressible mhd and navier stokes equations abstract this paper examines the long term dissipativity and unconditional non linear stability of time integration algorithms for the incompressible mhd equations the results apply as a particular case to the uncoupled incompressible navier stokes equations the analysis is performed in the setting defined by an abstract evolution equation possessing the same long term dissipative structure as the above physical systems a general class of time stepping schemes is analyzed in this context showing the non linear unconditional stability and long term dissipativity of particular members of this family of algorithms in particular algorithms showing these strong properties which additionally involve the solution of a linear problem at each time step are identified a particular linear algorithm is shown to exhibit optimal long term dissipative properties and to possess an algorithmic global attractor these abstract results apply rigorously to galerkin mixed finite element discretizations of the problem to this end it is shown that such galerkin type projection inherits the dissipative properties of the continuum problem computational aspects involved in the numerical implementation of these methods are addressed and illustrated in representative numerical simulations

discrete time lagrangian mechanics on lie groups with an application to the lagrange top we develop the theory of discrete time lagrangian mechanics on lie groups originated in the work of veselov and moser and the theory of lagrangian reduction in the discrete time setting the results thus obtained are applied to the investigation of an integrable time discretization of a famous integrable system of classical mechanics the lagrange top we recall the derivation of the euler poinsot equations of motion both in the frame moving with the body and in the rest frame the latter ones being less widely known we find a discrete time lagrange function turning into the known continuous time lagrangian in the continuous limit and elaborate both descriptions of the resulting discrete time system namely in the body frame and in the rest frame this system naturally inherits poisson properties of the continuous time system the integrals of motion being deformed the discrete time lax representations are also found kirchhoff s kinetic analogy between elastic curves and motions of the lagrange top is also generalised to the discrete context

thermodynamically consistent time stepping algorithms for non linear thermomechanical systems we present the basic theory for developing novel monolithic and staggered time stepping algorithms for general non linear coupled thermomechanical problems the proposed methods are thermodynamically consistent in the sense that their solutions rigorously comply with the two laws of thermodynamics for isolated systems they preserve the total energy and the entropy never decreases furthermore if the governing equations of the problem have symmetries the proposed integrators preserve them too the formulation of such methods is based on two ideas expressing the evolution equation in the so called general equations for non equilibrium reversible irreversible coupling format and enforcing from their inception certain directionality and degeneracy conditions on the discrete vector fields the new methods can be considered as an extension of the energy momentum integration algorithms to coupled thermomechanical problems to which they reduce in the purely hamiltonian case in the article the new ideas are applied to a simple coupled problem a double thermoelastic pendulum with symmetry numerical simulations verify the qualitative features of the proposed methods and illustrate their excellent numerical stability which stems precisely from their ability to preserve the structure of the evolution equations they discretize copyright 2009 john wiley sons ltd

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

discrete riemannian geometry within a framework of noncommutative geometry we develop an analog of pseudo riemannian geometry on finite and discrete sets on a finite set there is a counterpart of the continuum metric tensor with a simple geometric interpretation the latter is based on a correspondence between first order differential calculi and digraphs the vertices of the latter are given by the elements of the finite set arrows originating from a vertex span its co tangent space if the metric is to measure length and angles at some point it has to be taken as an element of the left linear tensor product of the space of 1 forms with itself and not as an element of the nonlocal tensor product over the algebra of functions as considered previously by several authors it turns out that linear connections can always be extended to this left tensor product so that metric compatibility can be defined in the same way as in continuum riemannian geometry in particular in the case of the universal differential calculus on a finite set the euclidean geometry of polyhedra is recovered from conditions of metric compatibility and vanishing torsion in our rather general framework which also comprises structures which are far away from continuum differential geometry there is in general nothing like a ricci tensor or a curvature scalar because of the nonlocality of tensor products over the algebra of functions of forms corresponding components with respect to some module basis turn out to be rather nonlocal objects but one can make use of the parallel transport associated with a connection to localize such objects and in certain cases there is a distinguished way to achieve this in particular this leads to covariant components of the curvature tensor which allow a contraction to a ricci tensor several examples are worked out to illustrate the procedure furthermore in the case of a differential calculus associated with a hypercubic lattice we propose a new discrete analogue of the vacuum einstein equations

a modified wavelet approximation for multi resolution awcm in simulating nonlinear vibration of mdof systems abstract wavelet methods have been presented recently to solve the initial value problems of time dependent odes before applying the current wavelet methods a nonlinear vibration equation usually needs to be transformed to state equations which may raise the storage and computation cost in this paper a modified form of wavelet approximation for the vibration solutions of linear nonlinear mdof systems is presented based on which a multi resolution awcm adaptive wavelet collocation method is established for mdof systems the modified wavelet approximation s wavelet coefficients explicitly include the boundary first order time derivatives of the solutions besides values of solutions on inner time locations of the time domain on which the solutions are defined this can avoid to transform the dynamical equations of a linear nonlinear vibrational mdof system to state equations therefore by using the modified wavelet approximation the computation and storage cost required for wavelet collocation method may be reduced to at most 1 2 of that needed by those based on state equations and the computation efficiency may be at least twice faster than wavelet collocation methods based on state equations because the number of solution components that are needed to be computed is only 1 2 of the corresponding state equations on the other hand the implementation of initial conditions is straightforward by using the modified wavelet approximation proposed in this paper the effectiveness of the modified wavelet approximation is tested by applying it to a linear and a nonlinear 10 dof vibrational system

complementary operators a method to annihilate artificial reflections arising from the truncation of the computational domain in the solution of partial differential equations the ease and simplicity with which the finite difference time domain fdtd or the finite elements fe techniques can handle complex radiation or scattering problems have lead to a remarkable surge in the use of these methods while execution time is becoming less of an impediment when solving large problems the biggest constraint remains the memory needed to run the fdtd or the fe methods it is precisely this limitation that the article addresses a boundary operation is developed to minimize the artificial reflections that arise when truncating the computational domain of an open region scattering or radiation problem the method is based on the use of two boundary operators that are complementary in their action by solving the problem with each of the two operators and then averaging the two solutions the first order reflections that arise from the artificial boundary can be completely eliminated numerical results are presented to show that this new technique gives significant reduction in the error when compared to other widely used boundary conditions

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

hamiltonian conserving discrete canonical equations based on variational difference quotients abstract new discrete mechanics based on the assumption of the discrete time is proposed the discrete mechanics does not contain any continuous differentiation but contains only difference quotients resulting discrete hamiltonian s canonical equations are single time step difference equations and exactly conserve the hamiltonian the canonical equations give the numerical results more accurately than the heun scheme and the 4th order runge kutta scheme

inherently energy conserving time finite elements for classical mechanics in this paper we develop a finite element method for the temporal discretization of the equations of motion the continuous galerkin method is based upon a weighted residual statement of hamilton s canonical equations we show that the proposed finite element formulation is energy conserving in a natural sense a family of implicit one step algorithms is generated by specifying the polynomial approximation in conjunction with the quadrature formula used for the evaluation of time integrals the numerical implementation of linear quadratic and cubic time finite elements is treated in detail for the model problem of a circular pendulum in addition to that concerning dynamical systems with several degrees of freedom we address the design of nonstandard quadrature rules which retain the energy conservation property our numerical investigations assess the effect of numerical quadrature in time on the accuracy and energy conservation property of the time stepping schemes

variational integrators and the newmark algorithm for conservative and dissipative mechanical systems the purpose of this work is twofold first we demonstrate analytically that the classical newmark family as well as related integration algorithms are variational in the sense of the veselov formulation of discrete mechanics such variational algorithms are well known to be symplectic and momentum preserving and to often have excellent global energy behavior this analytical result is veried through numerical examples and is believed to be one of the primary reasons that this class of algorithms performs so well second we develop algorithms for mechanical systems with forcing and in particular for dissipative systems in this case we develop integrators that are based on a discretization of the lagrange d alembert principle as well as on a variational formulation of dissipation it is demonstrated that these types of structured integrators have good numerical behavior in terms of obtaining the correct amounts by which the energy changes over the integration run

on the integrability of a representation of sl 2 r abstract the dunkl operators involve a multiplicity function k as parameter c f dunkl differential difference operators associated to reflection groups trans amer math soc 311 1989 167 183 for positive real values of this function we consider on the schwartz space s r n a representation k of sl 2 r defined in terms of the dunkl laplacian operator by means of a beautiful theorem due to e nelson we prove that k exponentiates to a unique unitary representation of the universal covering group of sl 2 r the representation theory is used to derive an identity of bochner type for the dunkl transform

discrete total variation calculus and lee s discrete mechanics a discrete total variation calculus is presented in this paper using this calculus we prove that the solution of the system of difference equations in lee s discrete mechanics preserves a symplectic structure in the space time sense numerical experiments are also reported

dunkl kernel associated with dihedral groups abstract in this paper we pursue the investigations started in 18 where the authors provide a construction of the dunkl intertwining operator for a large subset of the set of regular multiplicity values more precisely we make concrete the action of this operator on homogeneous polynomials when the root system is of dihedral type and under a mild assumption on the multiplicity function in particular we obtain a formula for the corresponding dunkl kernel and another representation of the generalized bessel function already derived in 7 when the multiplicity function is everywhere constant our computations give a solution to the problem of counting the number of factorizations of an element from a dihedral group into a fixed number of non necessarily simple reflections in the remainder of the paper we supply another method to derive the dunkl kernel associated with dihedral systems from the corresponding generalized bessel function this time we use the shift principle together with multiple combinations of dunkl operators in the directions of the vectors of the canonical basis of r 2 when the dihedral system is of order six and only in this case a single combination suffices to get the dunkl kernel and agrees up to an isomorphism with the formula recently obtained by amri 2 lemma 1 in the case of a root system of type a 2 we finally derive an integral representation for the dunkl kernel associated with the dihedral system of order eight

a variationally consistent mesh adaptation method for triangular elements in explicit lagrangian dynamics in this paper a variational formulation for mesh adaptation procedures involving local mesh changes for triangular meshes is presented such local adaptive changes are very well suited for explicit methods as they do not involve significant computational expense they also greatly simplify the projection of field variables from the old to the new meshes crucially the variational nature of the formulation used to derive the equilibrium equations at steps where adaptation takes place ensures that conservation of linear and angular momentum is obtained int j numer meth engng 2000 49 1295 1325 several examples in 2 d showing the application of the proposed adaptive algorithms are used to demonstrate the validity of the methodology proposed copyright 2009 john wiley sons ltd

a loop group approach to the c neumann problem and moser veselov factorization a geometrical description of continuous and discrete versions of the neumann oscillator in terms of a loop group framework is investigated it is shown that the continuous neumann oscillator can be integrated by the riemann hilbert factorization on the twisted loop group of o 3 lo 3 the solution of the problem is given in terms of a special class of flows on the quotient space lo 3 lo 3 where lo 3 is a subgroup of positive loops in lo 3 it is also shown that the moser veselov algorithm for integrating a discrete version of the neumann oscillator heisenberg xyz chain is induced by a discrete flow in this space the flow can be explicitly integrated by solving the matrix riccati equation in both cases discrete and continuous conservation laws are derived from time invariance of a relation that holds for coefficients of the fourier expansion of the flows

semi implicit reversible algorithms for rigid body rotational dynamics this paper presents two semi implicit algorithms based on splitting methodology for rigid body rotational dynamics the first algorithm is a variation of partitioned runge kutta prk methodology that can be formulated as a splitting method the second algorithm is akin to a multiple time stepping scheme and is based on modified crouch grossman mcg methodology which can also be expressed as a splitting algorithm these algorithms are second order accurate and time reversible however they are not poisson integrators i e non symplectic these algorithms conserve some of the first integrals of motion but some others are not conserved however the fluctuations in these invariants are bounded over exponentially long time intervals these algorithms exhibit excellent long term behavior because of their reversibility property and their approximate poisson structure preserving property the numerical results indicate that the proposed algorithms exhibit superior performance compared to some of the currently well known algorithms such as the simo wong algorithm newmark algorithm discrete moser veselov algorithm lewis simo algorithm and the liemid ea algorithm

geometric integrators and nonholonomic mechanics a geometric derivation of nonholonomic integrators is developed it is based in the classical technique of generating functions adapted to the special features of nonholonomic systems the theoretical methodology and the integrators obtained are different from those obtained in cortes and martinez nonholonomic integrators nonlinearity 14 1365 1392 2001 in the case of mechanical systems with linear constraints a family of geometric integrators preserving the nonholonomic constraints is given

parallel asynchronous variational integrators this paper presents a scalable parallel variational time integration algorithm for nonlinear elastodynamics with the distinguishing feature of allowing each element in the mesh to have a possibly different time step furthermore the algorithm is obtained from a discrete variational principle and hence it is termed parallel asynchronous variational integrator pavi the underlying variational structure grants it outstanding conservation properties based on a domain decomposition strategy pavi combines a careful scheduling of computations with fully asynchronous communications to provide a very efficient methodology for finite element models with even mild distributions of time step sizes numerical tests are shown to illustrate pavi s performance on both slow and fast networks showing scalability properties similar to the best parallel explicit synchronous algorithms with lower execution time finally a numerical example in which pavi needs 100 times less computing than an explicit synchronous algorithm is shown copyright 2006 john wiley sons ltd

variational integrators for forced birkhoffian systems in this letter we first extend the pfaff birkhoff principle to include forces and obtain the pfaff birkhoff d alembert principle consequently well then motions of forced birkhoffian systems coincide with extremals of the modified principle subsequently compared with the continuous case the discrete forced birkhoffian equations are constructed by discretizing the pfaff birkhoff d alembert principle considered as algorithms the discrete equations have good numerical behavior in terms of getting the correct amounts by which the energy changes over the integration process demonstrated by the given example

explicit formulas for the dunkl dihedral kernel and the κ a generalized fourier kernel abstract in this paper a new method is developed to obtain explicit and integral expressions for the kernel of the a generalized fourier transform for 0 in the case of dihedral groups this method is also applied to the dunkl kernel as well as the dunkl bessel function the method uses the introduction of an auxiliary variable in the series expansion of the kernel which is subsequently laplace transformed the kernel in the laplace domain takes on a much simpler form by making use of the poisson kernel the inverse laplace transform can then be computed using the generalized mittag leffler function to obtain integral expressions in case the parameters involved are integers explicit formulas are obtained using partial fraction decomposition new bounds for the kernel of the a generalized fourier transform are obtained as well

long run accuracy of variational integrators in the stochastic context this paper presents a lie trotter splitting for inertial langevin equations geometric langevin algorithm and analyzes its long time statistical properties the splitting is defined as a composition of a variational integrator with an ornstein uhlenbeck flow assuming that the exact solution and the splitting are geometrically ergodic the paper proves the discrete invariant measure of the splitting approximates the invariant measure of inertial langevin equations to within the accuracy of the variational integrator in representing the hamiltonian in particular if the variational integrator admits no energy error then the method samples the invariant measure of inertial langevin equations without error numerical validation is provided using explicit variational integrators with first second and fourth order accuracy

conservation properties of a time fe method part iv higher order energy and momentum conserving schemes in the present paper a systematic development of higher order accurate time stepping schemes which exactly conserve total energy as well as momentum maps of underlying finite dimensional hamiltonian systems with symmetry is shown the result of this development is the enhanced galerkin eg finite element method in time the conservation of the eg method is generally related to its collocation property total energy conservation in particular is obtained by a new projection technique the eg method is moreover based on objective time discretization of the used strain measure this paper is concerned with particle dynamics and semi discrete non linear elastodynamics the related numerical examples show good performance in presence of stiffness as well as for calculating large strain motions copyright 2005 john wiley sons ltd

algorithms for coupled problems that preserve symmetries and the laws of thermodynamics part i monolithic integrators and their application to finite strain thermoelasticity abstract we propose discretizations for a general class of nonlinear coupled thermomechanical problems of evolution the most salient feature of the new methods is that they rigorously preserve the two laws of thermodynamics as well as the symmetries of the systems they model to formulate such methods we exploit the geometric structure afforded by the generic formalism of non equilibrium thermodynamics and we follow a systematic methodology that results in discrete evolution equations which mimic the generic structure as an illustration a complete discretization of finite strain thermoelasticity is presented using finite elements in space and a monolithic integrator in time simulations are provided which demonstrate the conservation features of the algorithm as well as its remarkable robustness

a product formula and a convolution structure for a k hankel transform on r abstract the product b k x b k y of two dunkl type kernels is expressed as an integral in terms of b k z with explicit non positive kernel as a consequence a non positivity preserving convolution structure is established

asynchronous multi domain variational integrators for non linear problems we develop an asynchronous time integration and coupling method with domain decomposition for linear and non linear problems in mechanics to ensure stability in the time integration and in coupling between domains we use variational integrators with local lagrange multipliers to enforce continuity at the domain interfaces the asynchronous integrator lets each domain step with its own time step using a smaller time step where required by stability and accuracy constraints and a larger time step where allowed we show that in practice the time step is limited by accuracy requirements rather than by stability requirements copyright 2008 john wiley sons ltd

geometry and hamiltonian mechanics on discrete spaces numerical simulation is often crucial for analysing the behaviour of many complex systems which do not admit analytic solutions to this end one either converts a smooth model into a discrete in space and time model or models systems directly at a discrete level the goal of this paper is to provide a discrete analogue of differential geometry and to define on these discrete models a formal discrete hamiltonian structure in doing so we try to bring together various fundamental concepts from numerical analysis differential geometry algebraic geometry simplicial homology and classical hamiltonian mechanics for example the concept of a twisted derivation is borrowed from algebraic geometry for developing a discrete calculus the theory is applied to a nonlinear pendulum and we compare the dynamics obtained through a discrete modelling approach with the dynamics obtained via the usual discretization procedures also an example of an energy conserving algorithm on a simple harmonic oscillator is presented and its effect on the poisson structure is discussed

a time discontinuous implicit variational integrator for stress wave propagation analysis in solids this paper proposes a time discontinuous and space continuous variational integration tdsc vi method for accurate elastic stress wave propagation computations in one dimensional bar the algorithm employs a limiter akin to classical artificial viscosity treatment to mitigate the deleterious gibbs jumps across the stress discontinuities and a parametrized consistent mass to alleviate dispersion error when the stepsizes are different from the courant stability limit which becomes necessary for elastic unloading and internal reflections in plastic deformation problems stability and accuracy analyses of the proposed tdsc vi method are carried out and compared with several well known traditional integration algorithms numerical experiments are carried out with the proposed implicit and explicit methods which show that the proposed methods perform favorably compared to the trapezoidal rule and the central difference method

incursive discretization system bifurcation and energy conservation incursive discretization of the classical harmonic oscillator leads to system bifurcation the resulting hyperincursive representation has two alternative distinct algorithms of ordered serial noncommuting instructions and admits solutions having a discretized classical total energy that is perfectly conserved and phase space trajectories that are fully stable at all time scales hyperincursive representations can be generated for any hamiltonian system

discrete variational hamiltonian mechanics the main contribution of this paper is to present a canonical choice of a hamiltonian theory corresponding to the theory of discrete lagrangian mechanics we make use of lagrange duality and follow a path parallel to that used for construction of the pontryagin principle in optimal control theory we use duality results regarding sensitivity and separability to show the relationship between generating functions and symplectic integrators we also discuss connections to optimal control theory and numerical algorithms

rigid body dynamics in terms of quaternions hamiltonian formulation and conserving numerical integration in the present paper unit quaternions are used to describe the rotational motion of a rigid body the unit length constraint is enforced explicitly by means of an algebraic constraint correspondingly the equations of motion assume the form of differential algebraic equations daes a new route to the derivation of the mass matrix associated with the quaternion formulation is presented in contrast to previous works the newly proposed approach yields a non singular mass matrix consequently the passage to the hamiltonian framework is made possible without the need to introduce undetermined inertia terms the hamiltonian form of the daes along with the notion of a discrete derivative make possible the design of a new quaternion based energy momentum scheme two numerical examples demonstrate the performance of the newly developed method in this connection comparison is made with a quaternion based variational integrator a director based energy momentum scheme and a momentum conserving scheme relying on the discretization of the classical euler s equations

unitary integration with operator splitting for weakly dissipative systems unitary integration is a numerical method that preserves the structure of the quantum liouville equation by evolving the density via unitary transformations unitary integrators preserve the kinematic invariants cj tr j j 1 n to all orders in the time step here we extend unitary integration to weakly dissipative systems we apply the technique of operator splitting using a unitary integrator for the hamiltonian evolution and a conventional integrator for the dissipative piece in this way we guarantee that all dissipation and decoherence variation of the cj is due to the new non hamiltonian terms and not to any numerical artifacts we illustrate the method with examples

local path fitting a new approach to variational integrators a new approach for constructing variational integrators is presented in the general case the estimation of the action integral in a time interval t k t k 1 is used to construct a symplectic map q k q k 1 q k 1 q k 2 the basic idea is that only the partial derivatives of the estimated action integral of the lagrangian are needed in the general theory the analytic calculation of these derivatives gives rise to a new integral that depends on the euler lagrange vector itself which in the continuous and exact case vanishes and not on the lagrangian since this new integral can only be computed through a numerical method based on some internal grid points we can locally fit the exact curve by demanding the euler lagrange vector to vanish at these grid points thus the integral vanishes and the process dramatically simplifies the calculation of high order approximations the new technique is tested in high order solutions of the two body problem with high eccentricity up to 0 99 and of the outer planets of the solar system

asynchronous multi domain variational integrators for nonlinear hyperelastic solids we present the asynchronous multi domain variational time integrators with a dual domain decomposition method for the initial hyperbolic boundary value problem in hyperelasticity variational time integration schemes based on the principle of minimal action within the lagrangian framework are constructed for the equation of motion and implemented into a variational finite element framework which is systematically derived from the three field de veubeke hu washizu variational principle to accommodate the incompressibility constraint present in an analysis of nearly incompressible materials for efficient parallel computing we use the dual domain decomposition method with local lagrange multipliers to ensure the continuity of the displacement field at the interface between subdomains the method for time discretization and the multi domain spatial decomposition enable us to use different types of integrators explicit vs implicit and different time steps on different parts of a computational domain and thus efficiently capture the underlying physics with less computational effort the energy conservation of our nonlinear midpoint asynchronous integration scheme is investigated using the energy method and both local and the global energy error estimates are derived we illustrate the performance of proposed variational multi domain time integrators by means of three examples first the method of manufactured solutions is used to examine the consistency of the formulation in the second example we investigate energy conservation and stability finally we apply the method to the motion of a heterogeneous plane domain where different integrators and time discretization steps are used accordingly with disparate material data of individual parts

translation operator and maximal function for the k 1 generalized fourier transform abstract in this paper we study a translation operator associated with the n dimensional k 1 generalized fourier transform where k is a multiplicity function for the dunkl operators in particular we prove that the translation is a positivity preserving operator acting on a suitable space of radial functions on r n we then use it to define a hardy littlewood type maximal operator where weak type 1 1 and strong type p p estimates for the maximal operator are established with a precise behavior in n and k

symplectic numerical integrators in constrained hamiltonian systems recent work reported in the literature suggests that for the long time integration of hamiltonian dynamical systems one should use methods that preserve the symplectic or canonical structure of the flow here we investigate the symplecticness of numerical integrators for constrained dynamics such as occur in molecular dynamics when bond lengths are made rigid in order to overcome stepsize limitations due to the highest frequencies this leads to a constrained hamiltonian system of smaller dimension previous work has shown that it is possible to have methods which are symplectic on the constraint manifold in phase space here it is shown that the very popular verlet method with shake type constraints is equivalent to the same method with rattle type constraints and that the latter is symplectic and time reversible this assumes that the iteration is carried to convergence we also demonstrate the global convergence of the verlet scheme in the presence of shake type and rattle type constraints we perform numerical experiments to compare these methods with the second order backward differentiation method commonly recommended for ordinary differential equations with constraints

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

lie group variational integrators for the full body problem we develop the equations of motion for full body models that describe the dynamics of rigid bodies acting under their mutual gravity the equations are derived using a variational approach where variations are defined on the lie group of rigid body configurations both continuous equations of motion and variational integrators are developed in lagrangian and hamiltonian forms and the reduction from the inertial frame to a relative frame is also carried out the lie group variational integrators are shown to be symplectic to preserve conserved quantities and to guarantee exact evolution on the configuration space one of these variational integrators is used to simulate the dynamics of two rigid dumbbell bodies

finite element analysis of nonsmooth contact this work develops robust contact algorithms capable of dealing with complex contact situations involving several bodies with corners amongst the mathematical tools we bring to bear on the problem is nonsmooth analysis following clarke f h clarke optimization and nonsmooth analysis john wiley and sons new york 1983 we specifically address contact geometries for which both the use of normals and gap functions have difficulties and therefore precludes the application of most contact algorithms proposed to date such situations arise in applications such as fragmentation where angular fragments undergo complex collision sequences before they scatter we demonstrate the robustness and versatility of the nonsmooth contact algorithms developed in this paper with the aid of selected two and three dimensional applications

explicit newmark verlet algorithm for time integration of the rotational dynamics of rigid bodies we reformulate the traditional velocity based vector space newmark algorithm for the rotational dynamics of rigid bodies that is for the setting of the so 3 lie group we show that the most naive re write of the vector space algorithm possesses the properties of symplecticity and almost momentum conservation thus we obtain an explicit algorithm for rigid body dynamics that matches or exceeds performance of existing algorithms but which curiously does not seem to have been considered in the open literature so far copyright 2005 john wiley sons ltd

higher order discrete variational problems with constraints an interesting family of geometric integrators for lagrangian systems can be defined using discretizations of the hamilton s principle of critical action this family of geometric integrators is called variational integrators in this paper we derive new variational integrators for higher order lagrangian mechanical system subjected to higher order constraints from the discretization of the variational principles we show that our methods are automatically symplectic and in consequence with a very good energy behavior additionally the symmetries of the discrete lagrangian imply that momentum is conserved by the integrator moreover we extend our construction to variational integrators where the lagrangian is explicitly time dependent finally some motivating applications of higher order problems are considered in particular optimal control problems for explicitly time dependent underactuated systems and an interpolation problem on riemannian manifolds

exterior difference systems and invariance properties of discrete mechanics invariance properties describe the fundamental physical laws in discrete mechanics can those properties be described in a geometric way we investigate an exterior difference system called the discrete euler lagrange system whose solution has one to one correspondence with solutions of discrete euler lagrange equations and use it to define the first integrals the preservation of the discrete symplectic form along the discrete hamilton phase flows and the discrete noether s theorem is also described in the language of difference forms

conservation laws for nqc type difference equations this paper presents a classification of all three point conservation laws for a large class of integrable difference equations that has been described by nijhoff quispel and capel we show that every nonlinear equation from this class has at least two nontrivial conservation laws most of the conservation laws that are found are new

asynchronous variational contact mechanics an asynchronous variational method for simulating elastica in complex contact and impact scenarios is developed asynchronous variational integrators 1 avis are extended to handle contact forces by associating dierent time steps to forces instead of to spatial elements by discretizing a barrier potential by an innite sum of nested quadratic potentials these extended avis are used to resolve contact while obeying momentum and energy conservation laws a series of two and three dimensional examples illustrate the robustness and good energy behavior of the method

a no go theorem for the existence of a discrete action principle abstract in this paper we study the problem of the existence of a least action principle for invertible second order dynamical systems discrete in time and space we show that when the configuration space is finite and arbitrary state transitions are allowed a least action principle does not exist for such systems we dichotomize discrete dynamical systems with infinite configuration spaces into those of finite type for which this theorem continues to hold and those not of finite type for which it is possible to construct a least action principle we also show how to recover an action by restriction of the phase space of certain second order discrete dynamical systems we provide numerous examples to illustrate each of these results

symplectic algorithms based on the principle of least action and generating functions summary in this paper the principle of least action and generating functions are used to construct symplectic numerical algorithms for finite dimensional autonomous hamiltonian systems the approximate action is obtained by approximating the generalized coordinates and momentums by lagrange polynomials and performing gaussian quadrature based on the principle of least action and the requirements of a canonical transformation different types of symplectic algorithms have been constructed by choosing different types of independent variables at two ends of the time step the symmetric property of the four types of symplectic algorithms proposed in this paper is discussed and the exact linear stability domain for small m n and g is discussed the linear stability and precision of different types of symplectic algorithms are tested using numerical examples copyright 2011 john wiley sons ltd

applications of exterior difference systems to variations in discrete mechanics in discrete mechanics difference equations describe the fundamental physical laws and exhibit many geometric properties can these equations be obtained in a geometric way using some techniques in exterior difference systems we investigate the discrete variational problem as an application we give a positive answer to the above question for the discrete newton s euler lagrange and hamilton s equations

on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms we reconsider the problem of the hamiltonian interpolation of symplectic mappings following moser s scheme we prove that for any mapping e analytic and e close to the identity there exists an analytic autonomous hamiltonian system he such that its time one mapping he differs from e by a quantity exponentially small in 1 e this result is applied in particular to the problem of numerical integration of hamiltonian systems by symplectic algorithms it turns out that when using an analytic symplectic algorithm of orders to integrate a hamiltonian systemk one actually follows exactly namely within the computer roundoff error the trajectories of the interpolating hamiltonian he or equivalently of the rescaled hamiltonian ke e 1he which differs fromk but turns out to be e5 close to it special attention is devoted to numerical integration for scattering problems

pathwise accuracy and ergodicity of metropolized integrators for sdes metropolized integrators for ergodic stochastic differential equations sdes are proposed that 1 are ergodic with respect to the known equilibrium distribution of the sdes and 2 approximate pathwise the solutions of the sdes on finitetime intervals both these properties are demonstrated in the paper and precise strong error estimates are obtained it is also shown that the metropolized integrator retains these properties even in situations where the drift in the sde is nonglobally lipschitz and vanilla explicit integrators for sdes typically become unstable and fail to be ergodic 2009 wiley periodicals inc

nonholonomic mechanical systems with symmetry this work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of lagrangian mechanics and with a view to control theoretical applications the basic methodology is that of geometric mechanics applied to the formulation of lagrange d alembert generalizing the use of connections and momentum maps associated with a given symmetry group to this case we begin by formulating the mechanics of nonholonomic systems using an ehresmann connection to model the constraints and show how the curvature of this connection enters into lagrange s equations unlike the situation with standard configuration space constraints the presence of symmetries in the nonholonomic case may or may not lead to conservation laws however the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables this momentum equation which plays an important role in control problems involves parallel transport operators and is computed explicitly in coordinates an alternative description using a body reference frame relates part of the momentum equation to the components of the euler poincar e equations along those symmetry directions consistent with the constraints one of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not an example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard a modified version of the skateboard which uses momentum coupling for locomotion generation we construct a synthesis of the mechanical connection and the ehresmann connection defining the constraints obtaining an important new object we call the nonholonomic connection when the nonholonomic connection is a principal connection for the given symmetry group we show how to perform lagrangian reduction in the presence of nonholonomic constraints generalizing previous results which only held in special cases several detailed examples are given to illustrate the theory september 1994 revised march 1995 revised june 1995

an ergodic sampling scheme for constrained hamiltonian systems with applications to molecular dynamics this article addresses the problem of computing the gibbs distribu tion of a hamiltonian system that is subject to holonomic constraints in doing so we extend recent ideas of cances et al m2an vol 41 no 2 pp 351 389 2007 who could prove a law of large numbers for unconstrained molecular systems with a separable hamiltonian em ploying a discrete version of hamilton s principle studying ergodicity for constrained hamiltonian systems we specifically focus on the numerical discretization error even if the continuous system is perfectly ergodic this property is typically not preserved by the numerical discretization the discretization error is taken care of by means of a hybrid monte carlo al gorithm that allows for sampling bias free expectation values with respect to the gibbs measure independently of the stable step size we give a demonstration of the sampling algorithm by calculating the free energy profile of a small peptide

rattle a velocity version of the shake algorithm for molecular dynamics calculations an algorithm called rattle for integrating the equations of motion in molecular dynamics calculations for molecular models with internal constraints is presented the algorithm is similar to shake which is one of the standard methods for performing such calculations rattle calculates the positions and velocities at the next time from the positions and velocities at the present time step without requiring information about the earlier history like shake it is based on the verlet algorithm and retains the simplicity of using cartesian coordinates for each of the atoms to describe the configuration of a molecule with internal constraints rattle guarantees that the coordinates and velocities of the atoms in a molecule satisfy the internal constraints at each time step rattle has two advantages over shake on computers of fixed precision it is of higher precision than shake since it deals directly with the velocities it is easier to modify rattle for use with the recently developed constant temperature and constant pressure molecular dynamics methods and with the nonequilibrium molecular dynamics methods that make use of rescaling of the atomic velocities

a numerical integrator for the two fixed centres problem conserving all constants of motion the two fixed centres problem describes the motion of a particle influenced by the gravitational pull of two fixed particles this is a well known integrable hamiltonian system having three constants of motion this paper presents a discrete two dimensional two fixed centres problem which conserves two of these constants of motion moreover this discrete system preserves the discrete analogue of the third constant of motion a canonical transformation is introduced to separate the variables and remove singularities of the system a combination of this transformation and energy preserving method plays an important role in deriving the discrete system with a variable time step

error analysis of variational integrators of unconstrained lagrangian systems an error analysis of variational integrators is obtained by blowing up the discrete variational principles all of which have a singularity at zero time step divisions by the time step lead to an order that is one less than observed in simulations a deficit that is repaired with the help of a new past future symmetry

momentum conserving symplectic integrators abstract in this paper we show that symplectic partitioned runge kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of hamiltonian differential equations by extended point transformation we also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of lie poisson systems on certain submanifolds of the general matrix group gl n

mechanical integrators derived from a discrete variational principle many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion in this paper we present a procedure to construct time stepping algorithms that approximate the flow of continuous odes for mechanical systems by discretizing hamilton s principle rather than the equations of motion the discrete equations share similarities to the continuous equations by preserving invariants including the symplectic form and the momentum map we first present a formulation of discrete mechanics along with a discrete variational principle we then show that the resulting equations of motion preserve the symplectic form and that this formulation of mechanics leads to conservation laws from a discrete version of noether s theorem we then use the discrete mechanics formulation to develop a procedure for constructing symplectic momentum mechanical integrators for lagrangian systems with holonomic constraints we apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators

unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum we show that for rigid body dynamics the mid point rule formulated in body co ordinates exactly conserves energy and the norm of the angular momentum for incremental force free motions but fails to conserve the direction of the angular momentum vector further we show that the mid point rule formulated in the spatial representation is in general physically and geometrically meaningless an alternative algorithm is developed which exactly preserves energy and the total spatial angular momentum in incremental force free motions the implicit version of this algorithm is unconditionally stable and second order accurate the explicit version conserves exactly angular momentum in incremental force free motions numerical simulations are presented which illustrate the excellent performance of the proposed procedure even for incremental rotations over 65 degrees the procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells