long time estimates in the mean field limit in this paper we prove a stability result for measure perturbations of some class of stationary distributions of a vlasov equation we use this result to prove that the n particles approximation of these stationary distributions is uniformly valid on a time scale of order n1 8 which is much longer than the usual log n scale we also prove similar results for the approximation of the two dimensional euler equation by the vortex blob method

rational function approximations in the numerical solution of cauchy type singular integral equations cauchy type singular integral equations of the second kind with constant coefficients are solved via rational and polynomial approximations rational functions similar to that of polynomials have the property that for r t rational and for many of the weight functions w t encountered in practice r x 11w t r t t x dt is also rational hence approximations by rational functions is feasible rational function approximations in the solution of the dominant equation results in a linear algebraic system which possesses block diagonal structure it is further shown that the determinant of the coefficient matrix is bounded below away from zero and stability is ensured under fairly non restrictive conditions for the complete cauchy type singular integral equation i e the equation with both the principal and regular parts gaussian quadrature in conjunction with the rational function method is synthesized in the construction of a hybrid scheme error estimates and convergence are established a variety of problems from aerodynamics and fracture mechanics are solved and presented as a basis of comparison to polynomial based schemes

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

extended rearrangement inequalities and applications to some quantitative stability results in this paper we prove a new functional inequality of hardy littlewood type for generalized rearrangements of functions we then show how this inequality provides quantitative stability results of steady states to evolution systems that essentially preserve the rearrangements and some suitable energy functional under minimal regularity assumptions on the perturbations in particular this inequality yields a quantitative stability result of a large class of steady state solutions to the vlasov poisson systems and more precisely we derive a quantitative control of the l1 norm of the perturbation by the relative hamiltonian the energy functional and rearrangements a general non linear stability result has been obtained by lemou et al invent math 187 145 194 2012 in the gravitational context however the proof relied in a crucial way on compactness arguments which by construction provides no quantitative control of the perturbation our functional inequality is also applied to the context of 2d euler systems and also provides quantitative stability results of a large class of steady states to this system in a natural energy space

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

nonlinear stability criteria for the hmf model we study the nonlinear stability of a large class of inhomogeneous steady state solutions to the hamiltonian mean field hmf model under a specific criterion we prove the nonlinear stability of steady states which are decreasing functions of the microscopic energy to achieve this task we extend to this context the strategy based on generalized rearrangement techniques which was developed recently for the gravitational vlasov poisson equation explicit stability inequalities are established and our analysis is able to treat non compactly supported steady states to hmf which are physically relevant in this context but induces additional difficulties compared to the vlasov poisson system

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

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 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

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 geometry of non holonomic lagrangian systems we present a geometric framework for non holonomic lagrangian systems in terms of distributions on the configuration manifold if the constrained system is regular an almost product structure on the phase space of velocities is constructed such that the constrained dynamics is obtained by projecting the free dynamics if the constrained system is singular we develop a constraint algorithm which is very similar to that developed by dirac and bergmann and later globalized by gotay and nester special attention to the case of constrained systems given by connections is paid in particular we extend the results of koiller for caplygin systems an application to the so called non holonomic geometry is given

time discretized variational formulation of non smooth frictional contact the present work extends the non smooth contact class of algorithms introduced by kane et al to the case of friction the formulation specifically addresses contact geometries e g involving multiple collisions between tightly packed non smooth bodies for which neither normals nor gap functions can be properly defined a key aspect of the approach is that the incremental displacements follow from a minimum principle the objective function comprises terms which account for inertia strain energy contact friction and external forcing the euler lagrange equations corresponding to this extended variational principle are shown to be consistent with the equations of motion of solids in frictional contact in addition to its value as a basis for formulating numerical algorithms the variational framework offers theoretical advantages as regards the selection of trajectories in cases of non uniqueness we present numerical and analytical examples which demonstrate the good momentum and energy conservation characteristics of the numerical algorithms as well as the ability of the approach to account for stick and slip conditions

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

precise determination of the nonequilibrium tricritical point based on lynden bell theory in the hamiltonian mean field model existence of a nonequilibrium tricritical point has been revealed in the hamiltonian mean field model by a nonequilibrium statistical mechanics this statistical mechanics gives a distribution function containing unknown parameters and the parameters are determined by solving simultaneous equations depending on a given initial state due to difficulty in solving these equations pointwise numerical detection of the tricritical point has been unavoidable on a plane characterizing a family of initial states in order to look into the tricritical point we expand the simultaneous equations with respect to the order parameter and reduce them to one algebraic equation the tricritical point is precisely identified by analyzing coefficients of the reduced equation reentrance to an ordered phase in a high energy region is revisited around the obtained tricritical point

a variational technique for the solution of the singular integral equations over the real line using enriched spaces abstract mixed boundary value problems of mathematical physics may be formulated in a number of ways of interest is the numerical solution of those problems formulated as singular integral equations of cauchy type a variational technique is proposed for those equations defined over the real line i e ag x b g t t x dt k x t g t dt f x where a 0 b 1 and g k and f l 2 exploiting the basic property of the hilbert transform h i e h h an enriched space v e l 2 is constructed consisting of the chapeau functions and their images under the hilbert transform it is shown that besides being invariant under h v e is also complete in l 2 the solution g is bounded at the endpoints and is approximated in v e by a truncated fourier expansion the fourier coefficients are computed on orthonormalization of the images of the basis elements of v e the technique for the dominant equation and its extension to the complete equation is described convergence and the rate of convergence are established

towards a hamilton jacobi theory for nonholonomic mechanical systems in this paper we obtain a hamilton jacobi theory for nonholonomic mechanical systems the results are applied to a large class of nonholonomic mechanical systems the so called caplygin systems

spectral and formal stability criteria of spatially inhomogeneous stationary solutions to the vlasov equation for the hamiltonian mean field model stability of spatially inhomogeneous solutions to the vlasov equation is investigated for the hamiltonian mean field model to provide the spectral and formal stability criteria in the form of necessary and sufficient conditions these criteria determine stability of spatially inhomogeneous solutions whose stability has not been decided correctly by using a less refined formal stability criterion it is shown that some of such solutions can be found in a family of stationary solutions to the vlasov equation which is parametrized with macroscopic quantities and has a two phase coexistence region in the parameter space

orbital stability of spherical galactic models we consider the three dimensional gravitational vlasov poisson system which is a canonical model in astrophysics to describe the dynamics of galactic clusters a well known conjecture is the stability of spherical models which are nonincreasing radially symmetric steady states solutions this conjecture was proved at the linear level by several authors in the continuation of the breakthrough work by antonov in 1961 in a previous work we derived the stability of anisotropic models under it spherically symmetric perturbations using fundamental monotonicity properties of the hamiltonian under suitable generalized symmetric rearrangements first observed in the physics litterature in this work we show how this approach combined with a it new generalized antonov type coercivity property implies the orbital stability of spherical models under general perturbations

on the neighborhood of an inhomogeneous stable stationary solution of the vlasov equation case of an attractive cosine potential we consider the one dimensional vlasov equation with an attractive cosine potential and its non homogeneous stable stationary states that are decreasing functions of the energy we show that in the sobolev space w1 p p 2 neighborhood of such a state all stationary states that are decreasing functions of the energy are stable this is in sharp contrast with the situation for homogeneous stationary states of a vlasov equation where a control over strictly more than one derivative is needed to ensure the absence of unstable stationary states in a neighborhood of a reference stationary state z lin and c zeng commun math phys 306 291 331 2011

reduction of constrained mechanical systems and stability of relative equilibria a mechanical system with perfect constraints can be described under some mild assumptions as a constrained hamiltonian system m h d w m thephase space is a symplectic manifold h thehamiltonian a smooth function onm d theconstraint submanifold a submanifold ofm andw theprojection bundle a vector sub bundle oft d m the reduced tangent bundle alongd we prove that when these data satisfy some suitable conditions the time evolution of the system is governed by a well defined differential equation ond we define constrained hamiltonian systems with symmetry and prove a reduction theorem application of that theorem is illustrated on the example of a convex heavy body rolling without slipping on a horizontal plane two other simple examples show that constrained mechanical systems with symmetry may have an attractive or repulsive set of relative equilibria

reduction of some classical non holonomic systems with symmetry two types of nonholonomic systems with symmetry are treated i the configuration space is a total space of a g principal bundle and the constraints are given by a connection ii the configuration space is g itself and the constraints are given by left invariant forms the proofs are based on the method of quasicoordinates in passing a derivation of the maurer cartan equations for lie groups is obtained simple examples are given to illustrate the algorithmical character of the main results

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

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