symplectic integrators for index one constraints we show that symplectic runge kutta methods provide effective symplectic integrators for hamiltonian systems with index one constraints these include the hamiltonian description of variational problems subject to position and velocity constraints nondegenerate in the velocities such as those arising in sub riemannian geometry and control theory

the midpoint scheme and variants for hamiltonian systems advantages and pitfalls the implicit midpoint scheme like higher order gauss collocation schemes is algebraically stable and symplectic and it preserves quadratic integral invariants it may appear particularly suitable for the numerical solution of highly oscillatory hamiltonian systems such as those arising in molecular dynamics or structural mechanics because there is no stability restriction when it is applied to a simple harmonic oscillator although it is well known that the midpoint scheme may also exhibit instabilities in various stiff situations one might still hope for good results when resonance type instabilities are avoided in this paper we investigate the suitability of the midpoint scheme for highly oscillatory frictionless mechanical systems where the step size k is much larger than the system s small parameter varepsilon in the case that the solution remains bounded as varepsilon rightarrow 0 we show that in general one must require that k 2 varepsilon be small enough or else even the errors in slowly varying quantities like the energy may grow undesirably especially when fast and slow modes are tightly coupled or worse the computation may yield misleading information in some cases this may already happen when k o varepsilon the same holds for higher order collocation at gaussian points the encountered restrictions on k are typically still better than the corresponding ones for explicit schemes

multi multiplier ambient space formulations of constrained dynamical systems with an application to elastodynamics various formulations of the equations of motion for both finite and infinite dimensional constrained lagrangian dynamical systems are studied the different formulations correspond to different ways of enforcing constraints through multiplier fields all the formulations considered are posed on ambient spaces whose members are unrestricted by the need to satisfy constraint equations but each formulation is shown to possess an invariant set on which the constraint equations and physical balance laws are satisfied the stability properties of the invariant set within its ambient space are shown to be different in each case we use the specific model problem of linearized incompressible elastodynamics to compare properties of three different ambient space formulations we establish the well posedness of one formulation in the particular case of a homogeneous isotropic body subject to specified tractions on its boundary

langevin dynamics with constraints and computation of free energy differences in this paper we consider langevin processes with mechanical constraints the latter are a fundamental tool in molecular dynamics simulation for sampling purposes and for the computation of free energy differences the results of this paper can be divided into three parts i we propose a simple discretization of the constrained langevin process based on a standard splitting strategy we show how to correct the scheme so that it samples em exactly the canonical measure restricted on a submanifold using a metropolis rule in the spirit of the generalized hybrid monte carlo ghmc algorithm moreover we obtain in some limiting regime a consistent discretization of the overdamped langevin brownian dynamics on a submanifold also sampling exactly the correct canonical measure with constraints the corresponding numerical methods can be used to sample without any bias a probability measure supported by a submanifold ii for free energy computation using thermodynamic integration we rigorously prove that the longtime average of the lagrange multipliers of the constrained langevin dynamics yields the gradient of a rigid version of the free energy associated with the constraints a second order time discretization using the lagrange multipliers is proposed iii the jarzynski crooks fluctuation relation is proved for langevin processes with mechanical constraints evolving in time an original numerical discretization without time step error is proposed numerical illustrations are provided for ii and iii

the nose poincare method for constant temperature molecular dynamics we present a new extended phase space method for constant temperature canonical ensemble molecular dynamics our starting point is the hamiltonian introduced by nose to generate trajectories corresponding to configurations in the canonical ensemble using a poincare time transformation we construct a hamiltonian system with the correct intrinsic timescale and show that it generates trajectories in the canonical ensemble our approach corrects a serious deficiency of the standard change of variables nose hoover dynamics which yields a time reversible system but simultaneously destroys the hamiltonian structure a symplectic discretization method is presented for solving the nose poincare equations the method is explicit and preserves the time reversal symmetry in numerical experiments it is shown that the new method exhibits enhanced stability when the temperature fluctuation is large extensions are presented for nose chains holonomic constraints and rigid bodies

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

mechanical systems subject to holonomic constraints differential algebraic formulations and conservative integration abstract the numerical integration in time of the equations of motion for mechanical systems subject to holonomic constraints is considered schemes are introduced for the direct treatment of a differential algebraic form of the equations that preserve the constraints the total energy and other integrals such as linear and angular momentum arising from affine symmetries moreover the schemes can be shown to preserve the property of time reversibility in an appropriate sense an example is given to illustrate various aspects of the proposed methods

a time reversible variable stepsize integrator for constrained dynamics this article considers the design and implementation of variable timestep methods for simulating holonomically constrained mechanical systems symplectic variable stepsizes are briefly discussed and we consider time reparameterization techniques employing a time reversible symmetric integration method to solve the equations of motion we give several numerical examples including a simulation of an elastic inextensible unshearable rod undergoing large deformations and collisions with the sides of a bounding box numerical experiments indicate that adaptive stepping can significantly smooth the numerical energy and improve the overall efficiency of the simulation

on higher order semi explicit symplectic partitioned runge kutta methods for constrained hamiltonian systems in this paper we generalize the class of explicit partitioned runge kutta prk methods for separable hamiltonian systems to systems with holonomic constraints for a convenient analysis of such schemes we first generalize the backward error analysis for systems in bbb r m to systems on manifolds embedded in bbb r m by applying this analysis to constrained prk methods we prove that such methods will in general suffer from order reduction as well known for higher index differential algebraic equations however this order reduction can be avoided by a proper modification of the standard prk methods this modification increases the number of projection steps onto the constraint manifold but leaves the number of force evaluations constant we also give a numerical comparison of several second fourth and sixth order methods

computing global orbits of the forced spherical pendulum abstract orbits of the spherical pendulum with time periodic forcing are considered a numerical framework is developed which allows orbits to explore the entire globe the spherical pendulum is considered as an invariant manifold in an ambient six dimensional euclidean space the numerical integrator is the second order stormer verlet method coupled with the shake rattle algorithm the algorithm preserves numerically the phase space of the sphere which is a manifold to machine accuracy poincare sections restricted to the configuration space are used to illustrate the transition from oscillatory behavior to chaotic solutions as the amplitude of the pivot motion is changed the qualitative change in the poincare sections from regular to chaotic behavior is in excellent qualitative agreement with corresponding computations of the lyapunov exponents les the les are also computed using a novel variant of the shake rattle algorithm the results show that irregular behavior can explore the entire sphere even at low forcing amplitudes and therefore local methods which parameterize only part of the sphere are inadequate in general and may lead to spurious dynamics the numerical framework provides a tool for detailed investigation of the symmetric chaos of the forced spherical pendulum

wiggle a new constrained molecular dynamics algorithm in cartesian coordinates the theory of conventional constrained molecular dynamics md simulations is reexamined based on a projection operator approach a new algorithm named wiggle is presented for md simulations with internal constraints at each time step the algorithm utilizes the constrained accelerations derived from velocity adjustments to satisfy the hidden constraints and provides both initial and final constrained values that are almost an order of magnitude closer to the desired values than does rattle its performance is compared with those of rattle and shake for an octane molecule also presented are a formalism to additionally constrain the angular momentum about the center of mass and an expression for the local energy drift during each integration time step

stability and uniqueness properties of taylor approximations of matrix functions abstract we consider taylor approximations of matrix functions and establish bounds controlling the size of a perturbation in terms of the size of the taylor remainder or equivalently the associated spectral shift function s this can be viewed as a stability property for the inverse problem of recovering the perturbation from the spectral shift function s from our bounds we derive a uniqueness result for spectral sums if the trace of the taylor remainder of order n 2 equals zero for a finite number of monomials then the perturbation is zero this uniqueness result is in contrast with nonuniqueness of the first order taylor remainder as an application we characterize equality of two graphs in terms of self returning walks

suppressing local particle oscillations in the hamiltonian particle method for elasticity the governing equation of elasticity is discretized into motion equations of the particles in a hamiltonian system a weighted least square method is adopted to evaluate the green lagrange strain using a symplectic scheme for the hamiltonian system we obtain the property of energy conservation in the discretized calculations however local particle oscillations occur and they excessively decrease low frequency motion in this study we propose the use of an artificial potential force to suppress the local oscillations the accuracy of the model with and without the inclusion of the artificial force is examined by analyzing a cantilever beam and wave propagation with the inclusion of the artificial force the local oscillations are reduced while energy conservation is maintained

free versus constrained evolution of the 2 1 equivariant wave map we compare the numerical solutions of the 2 1 equivariant wave map problem computed with the symplectic constraint respecting rattle algorithm and the well known fourth order runge kutta method we show the advantages of the rattle algorithm for the constrained system compared to the free evolution with the runge kutta method we also present an expression which represents the energy loss due to constraint violation taking this expression into account we can achieve energy conservation for the runge kutta scheme which is better than with the rattle method using the symplectic scheme with constraint enforcement we can reproduce previous calculations of the equivariant case without imposing the symmetry explicitly thereby confirming that the critical behaviour is stable

smoothed langevin dynamics of highly oscillatory systems abstract we consider hamiltonian systems containing a strong constraining potential such systems were first investigated by rubin and ungar who derived the corresponding reduced equations of motion for the slow degree of freedom the purpose of this paper is two fold i we transform the highly oscillatory degree of freedom to action angle variables and use recent results on the truncation error of the corresponding normal form transformations to derive an estimate on the lifespan of the underlying adiabatic invariant ii we investigate thermally embedded hamiltonian systems i e langevin dynamics containing a strong constraining potential using again a transformation to action angle variables and applying the principle of averaging we derive reduced equations of motion for the slowly varying degrees of freedom this includes in particular the derivation of a correcting force term that stands for the coupling of the slow and fast degrees of motion in case of strong thermal coupling our reduced equations of motion suggest that the correcting force term can be approximated by the gradient of the fixman potential as used in statistical mechanics we also discuss smoothing in the context of constant temperature molecular dynamics and provide a numerical example

a patch that imparts unconditional stability to explicit integrators for langevin like equations this paper extends the results in 8 to stochastic differential equations sdes arising in molecular dynamics it implements a patch to explicit integrators that consists of a metropolis hastings step the patched integrator preserves the sde s equilibrium distribution and is accurate on finite time intervals as a corollary this paper proves the integrator s accuracy in estimating finite time dynamics along an infinitely long solution a first in molecular dynamics the paper also covers multiple time steps holonomic constraints and scalability finally the paper provides numerical tests supporting the theory

symplectic integration of constrained hamiltonian systems a hamiltonian system in potential form h q p ptmp 2 f q subject to smooth constraints on q can be viewed as a hamiltonian system on a manifold but numerical computations must be performed in rn in this paper methods which reduce hamiltonian differential algebraic equations to odes in euclidean space are examined the authors study the construction of canonical parametrizations or local charts as well as methods based on the construction of ode systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an invariant manifold in each case a hamiltonian system of ordinary differential equations is produced the stability of the constraint invariants and the behavior of the original hamiltonian along solutions are investigated both numerically and analytically

conservation properties of a time fe method part iii mechanical systems with holonomic constraints a galerkin based discretization method for index 3 differential algebraic equations pertaining to finite dimensional mechanical systems with holonomic constraints is proposed in particular the mixed galerkin mg methodis introduced which leads in a natural way to time stepping schemes that inherit major conservation properties of the underlying constrained hamiltonian system namely total energy and angular momentum in addition to that the constraints on the configuration level and on the velocity momentum level are fulfilled exactly the application of the mg method to specific mechanical systems such as the pendulum rigid body dynamics and the coupled motion of rigid and flexible bodies is presented related numerical examples are investigated to evaluate the numerical performance of the mg 1 and mg 2 method

efficient molecular dynamics using geodesic integration and solvent solute splitting we present an approach to langevin dynamics in the presence of holonomic constraints based on decomposition of the system into components representing geodesic flow constrained impulse and constrained diffusion we show that a particular ordering of the components results in an integrator that is an order of magnitude more accurate for configurational averages than existing alternatives moreover by combining the geodesic integration method with a solvent solute force splitting we demonstrate that stepsizes of at least 8 fs can be used for solvated biomolecules with high sampling accuracy and without substantially altering diffusion rates approximately increasing by a factor of two the efficiency of molecular dynamics sampling for such systems the methods described in this article are easily implemented using the standard apparatus of modern simulation codes

numerical integration of constrained hamiltonian systems using dirac brackets we study the numerical properties of the equations of motion of constrained systems derived with dirac brackets this formulation is compared with one based on the extended hamiltonian as concrete examples a pendulum in cartesian coordinates and a chain molecule are treated

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

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

hamiltonian moving particle semi implicit hmps method for incompressible fluid flows particle methods are meshless simulation techniques in which motion of continua is approximated by discrete dynamics of a finite number of particles and thus have a great degree of flexibility for instance in dealing with the complex motion of surfaces or boundaries in this paper a particle method is developed from a direct discretization of the lagrangian for inviscid incompressible fluid flows the discretized lagrangian is transformed into a hamiltonian with holonomic constraints that stem from the incompressibility condition the rattle algorithm that is a symplectic scheme for holonomically constrained hamiltonian systems is adopted so that the resulting particle method inherits structures possessed by the partial differential equations that govern inviscid incompressible flows the algorithm of the particle method is semi implicit so we call it the hamiltonian moving particle semi implicit hmps method some numerical tests indicate the excellence of the hmps method in conservation of mechanical energy as well as linear and angular momenta

a numerical scheme for optimal transition paths of stochastic chemical kinetic systems we present a new framework for finding the optimal transition paths of metastable stochastic chemical kinetic systems with large system size the optimal transition paths are identified to be the most probable paths according to the large deviation theory of stochastic processes dynamical equations for the optimal transition paths are derived using the variational principle a modified minimum action method mam is proposed as a numerical scheme to solve the optimal transition paths applications to gene regulatory networks such as the toggle switch model and the lactose operon model in escherichia coli are presented as numerical examples

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

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

sph simulations of swimming linked bodies in this paper we describe how the swimming of linked rigid bodies can be simulated using smoothed particle hydrodynamics sph the fluid is assumed to be viscous and weakly compressible though with a speed of sound which ensures the mach number is 0 1 and the density fluctuation relative to the average density is typically 0 01 the motion is assumed to be two dimensional the boundaries of the rigid bodies are replaced by boundary particles and the forces between these particles and the fluid particles determine the forces and torques on the rigid bodies the links between the bodies are described by constraint equations which are taken into account by lagrange multipliers we integrate the equations by a second order method which conserves linear momentum exactly and in the absence of viscosity is reversible we test our method by simulating the motion of a cylinder moving in a viscous fluid both under forced oscillation and when tethered to a spring we then apply our method to three problems involving three linked bodies the first of these consists of three linked diamonds in a periodic domain and shows the convergence of the algorithm the second is the system considered by kanso et al e kanso j e marsden c w rowley j b melli huber locomotion of articulated bodies in a perfect fluid j nonlinear sci 15 2005 255 289 referred to in the text as k05 and our results show similar behaviour for the same gait the third example clarifies the conservation of angular momentum by simulating the motion of three linked ellipses moving within and through the surface of an initially circular patch of fluid our method can be easily extended to linked bodies swimming near surfaces or to swimming near fixed boundaries of arbitrary shape or to swimming in fluids which are stratified some of these systems have biological significance and others are applicable to the study of undersea vessels which move because of shape changes

structure preservation for constrained dynamics with super partitioned additive runge kutta methods a broad class of partitioned differential equations with possible algebraic constraints is considered including hamiltonian and mechanical systems with holonomic constraints for mechanical systems a formulation eliminating the coriolis forces and closely related to the euler lagrange equations is presented a new class of integrators is defined the super partitioned additive runge kutta spark methods this class is based on the partitioning of the system into different variables and on the splitting of the differential equations into different terms a linear stability and convergence analysis of these methods is given spark methods allowing the direct preservation of certain properties are characterized different structures and invariants are considered the manifold of constraints symplecticness reversibility contractivity dilatation energy momentum and quadratic invariants with respect to linear stability and structure preservation the class of s stage lobatto iiia b c c spark methods is of special interest controllable numerical damping can be introduced by the use of additional parameters some issues related to the implementation of a reversible variable stepsize strategy are discussed

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

the discrete null space method for the energy consistent integration of constrained mechanical systems part i holonomic constraints the present work deals with energy consistent time stepping schemes for finite dimensional mechanical systems with holonomic constraints the proposed procedure is essentially based upon the following steps firstly the index three differential algebraic equations corresponding to the constrained mechanical system are directly discretized secondly the discrete lagrange multipliers are eliminated by using a discrete null space matrix in many cases it is feasible to further reduce the number of unknowns by employing specific reparametrizations the proposed method entails a number of advantageous features such as size reduction and improved conditioning of the resulting system of algebraic equations it is shown that the newly developed method is well suited for both open loop and closed loop multibody systems