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

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

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

unstable eigenvalues and the linearization about solitary waves and fronts with symmetry the linear stability of solitary wave or front solutions of hamiltonian evolutionary equations which are equivariant with respect to a lie group is studied the organizing centre for the analysis is a multisymplectic formulation of hamiltonian pdes where a distinct symplectic operator is assigned for time and space this separation of symplectic structures leads to new characterizations of the following components of the analysis the states at infinity are characterized as manifolds of relative equilibria associated with the spatial symplectic structure the momentum of the connecting orbit or shape of the solitary wave considered as a heteroclinic orbit in a phase space representation is given a new characterization as a one form on the tangent space to the heteroclinic manifold and this one form is a restriction of the temporal symplectic structure for the linear stability analysis a new symplectic characterization of the evans function and its derivatives are obtained leading to an abstract geometric proof of instability for a large class of solitary wave states of equivariant hamiltonian evolutionary pdes the theory sheds new light on several well known models the gkdv equation a boussinesq system and a nonlinear wave equation the generalization to solitary waves associated with multidimensional heteroclinic manifolds and the implications for solitary waves or fronts which are biasymptotic to invariant manifolds such as periodic states are also discussed

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

a newton method for rigid body frictional impact with multiple simultaneous impact points in this paper a mathematical formulation and a numerical algorithm for the analysis of frictional impact of rigid bodies with multiple simultaneous contact points are developed a direct approach is used where the impenetrability condition and coulomb s law of friction are formulated as equations which are not differentiable in the usual sense and solved together with the equations of motion and necessary kinematical relations using newton s method an experiment has also been performed where a possible mechanism for the difference between static and dynamic friction is observed and results are compared with computations using the present algorithm

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

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

on zariski s cancellation problem in positive characteristic abstract in this paper we show that when k is a field of positive characteristic the affine space a k n is not cancellative for any n 3

total variation in hamiltonian formalism and symplectic energy integrators we present a discrete total variation calculus in hamiltonian formalism in this paper using this discrete variation calculus and generating functions for the flows of hamiltonian systems we derive symplectic energy integrators of any finite order for hamiltonian systems from a variational perspective the relationship between the symplectic integrators derived directly from the hamiltonian systems and the variationally derived symplectic energy integrators is explored

derivation of lagrangian density for the good boussinesq equation and multisymplectic disretizations abstract the lagrangian density for the good boussinesq equation is derived by introducing a new variable in this paper based on this density we consider the corresponding multisymplectic formulation and discretizations numerical experiments are also reported

efficient algorithms for computing a strong rank revealing qr factorization given anm n matrixm withm n it is shown that there exists a permutation fi and an integer k such that the qr factorization myi q ak ckbk reveals the numerical rank of m the k k upper triangular matrix ak is well conditioned ilckll2 is small and bk is linearly dependent on ak with coefficients bounded by a low degree polynomial in n existing rank revealing qr rrqr algorithms are related to such factorizations and two algorithms are presented for computing them the new algorithms are nearly as efficient as qr with column pivoting for most problems and take o ran2 floating point operations in the worst case

formulation and comparison of algorithms for frictional contact problems this paper presents two algorithms for solving the discrete quasi static small displacement linear elastic contact problem with coulomb friction the algorithms are adoptions of a newton method for solving b differentiable equations and an interior point method for solving smooth constrained equations for the application of the former method the contact problem is formulated as a system of b differentiable equations involving the projection operator onto sets with simple structure for the application of the latter method the contact problem is formulated as a system of smooth equations involving complementarity conditions and with the non negativity of variables treated as constraints the two algorithms are numerically tested for two dimensional problems containing up to 100 contact nodes and up to 100 time increments results show that at the present stage of development the newton method is superior both in robustness and speed additional comparison is made with a commercial finite element code

a 3 d discrete element method for dry granular flows of ellipsoidal particles abstract we use 3 d discrete element method dem to simulate dry granular flows of non spherical particles in our simulation code non spherical particles are represented by clusters of spheres for contact detection and for contact force calculation for particles in contact the normal force displacement nfd model is simulated by a partially latching spring system whereas the tangential force displacement tfd model is simulated by an incremental relation that corresponds to a simplification of the mindlin and deresiewicz r d mindlin h deresiewicz asme j appl mech 20 1953 327 344 contact mechanics theory to establish the validity of the simulation procedure benchmark tests such as superball and superbean simulations were carried out we present various results obtained from the simulation of ellipsoidal particles flowing down a chute having a bumpy bottom these ellipsoidal particles had their physical properties obtained from experiments

on double danielewski surfaces and the cancellation problem abstract we study a two dimensional family of affine surfaces which are counter examples to the cancellation problem we describe the makar limanov invariant of these surfaces determine their isomorphism classes and characterize the automorphisms of these surfaces

a multisymplectic integrator for the periodic nonlinear schrodinger equation a multisymplectic integrator for the periodic nonlinear schrodinger equation is presented in this paper its accuracy is proved by introducing a norm we investigate its nonlinear stability we also discuss the relationship between this multisymplectic integrator and two variational integrators which are derived by using the discrete multisymplectic field theory and the finite element method

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

on the cancellation problem for the affine space mathbb a 3 in characteristic p we show that the cancellation conjecture does not hold for the affine space mathbb a 3 k over any field k of positive characteristic we prove that an example of t asanuma provides a three dimensional k algebra a for which a is not isomorphic to k x1 x2 x3 although a t is isomorphic to k x1 x2 x3 x4

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

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