Search results for: euler-lagrange equations
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Existence of Two Periodic Solutions to General Anisotropic Euler-Lagrange Equations
PublicationAbstract. This paper is concerned with the following Euler-Lagrange system d/dtLv(t,u(t), ̇u(t)) =Lx(t,u(t), ̇u(t)) for a.e.t∈[−T,T], u(−T) =u(T), Lv(−T,u(−T), ̇u(−T)) =Lv(T,u(T), ̇u(T)), where Lagrangian is given by L=F(t,x,v) +V(t,x) +〈f(t),x〉, growth conditions aredetermined by an anisotropic G-function and some geometric conditions at infinity.We consider two cases: with and without forcing termf. Using a general version...
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Mountain pass solutions to Euler-Lagrange equations with general anisotropic operator
PublicationUsing the Mountain Pass Theorem we show that the problem \begin{equation*} \begin{cases} \frac{d}{dt}\Lcal_v(t,u(t),\dot u(t))=\Lcal_x(t,u(t),\dot u(t))\quad \text{ for a.e. }t\in[a,b]\\ u(a)=u(b)=0 \end{cases} \end{equation*} has a solution in anisotropic Orlicz-Sobolev space. We consider Lagrangian $\Lcal=F(t,x,v)+V(t,x)+\langle f(t), x\rangle$ with growth conditions determined by anisotropic G-function and some geometric conditions...
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Mountain pass type periodic solutions for Euler–Lagrange equations in anisotropic Orlicz–Sobolev space
PublicationUsing the Mountain Pass Theorem, we establish the existence of periodic solution for Euler–Lagrange equation. Lagrangian consists of kinetic part (an anisotropic G-function), potential part and a forcing term. We consider two situations: G satisfying at infinity and globally. We give conditions on the growth of the potential near zero for both situations.
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Anisotropic Orlicz–Sobolev spaces of vector valued functions and Lagrange equations
PublicationIn this paper we study some properties of anisotropic Orlicz and Orlicz–Sobolev spaces of vector valued functions for a special class of G-functions. We introduce a variational setting for a class of Lagrangian Systems. We give conditions which ensure that the principal part of variational functional is finitely defined and continuously differentiable on Orlicz–Sobolev space.
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Adaptation of the arbitrary Lagrange–Euler approach to fluid–solid interaction on an example of high velocity flow over thin platelet
PublicationThe aim of this study is to analyse the behaviour of a thin plate with air flow velocities of 0.3–0.9 Ma. Data from the experiment and numerical tools were used for the analysis. For fluid–solid interaction calculations, the arbitrary Lagrange–Euler approach was used. The results of the measurements are twofold. The first one is the measurement of the flow before and after vibrating plate, i.e. pure flow plate, and the second consists...
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Generalized Euler method for first order partial differential functional equations
PublicationW pracy prezentowana jest nowa klasa metod numerycznych dla nieliniowych równań różniczkowo funkcyjnych pierwszego rzędu.Rozwiązania klasyczne zagadnień początkowo brzegowych przybliżane są w tej pracy przez rozwiązania odpowiedniego układu quasilininowego równań różnicowych. Podajemy kompletną analizę zbieżności metod i pokazujemy na przykładach, iż nowa metoda jest zauważalnie lepsza niż klasyczne schematy różnicowe. Dowód stabilności...
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Generalized Euler method for nonlinear first order partial differential equations.
PublicationKlasyczne rozwiązania nieliniowych równań różniczkowych cząstkowych pierwszego rzędu są aproksymowane w tej pracy za pomocą rozwiązań quasiliniowych układów równań różnicowych. Podstawowa idea pracy jest oparta na teorii charakterystyk. Podane są warunki wystarczające dla zbieżności metody. Dowód stabilności schematu różnicowego wykorzystuje metodę porównawczą z nieliniowymi oszacowaniami typu Perrona dla danych funkcji.Podane...
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Non-linearity of multibody dynamic equations with respect to Lagrange multipliers: application to railway dynamics
PublicationPraca koncentruje się na dynamice układów wieloczłonowych z zamkniętymi łańcuchami członów. Głównym punktem zainteresowania jest modelowanie układów z występującymi nieliniowymi zależnościami opisującymi wpływ siły mnożników Lagrange'a na dynamikę układu (nieliniowe modele siły tarcia.). Aby zbudować model dynamiki układu zawierającego zamknięte łańcuchy członów, wspomniane łańcuchy są "rozcinane" i budowana jest struktura drzewa...
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Approximate solution for Euler equations of stratified water via numerical solution of coupled KdV system
PublicationWe consider Euler equations with stratified background state that is valid for internal water waves. The solution of the initial-boundary problem for Boussinesq approximation in the waveguide mode is presented in terms of the stream function. The orthogonal eigenfunctions describe a vertical shape of the internal wave modes and satisfy a Sturm-Liouville problem. The horizontal profile is defined by a coupled KdV system which is...
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Variational principles for bound states of Schrödinger and Dirac equations allowing the use of discontinuous trial functions
PublicationWe present systematic constructions of variational principles for energies of bound states of the Schroedinger and Dirac equations. The principles allow the use of discontinuous trial functions. The method employed is based on a generalized Lagrange procedure. Relationships between our variational principles and those available in the literature are established.
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Closed Form Constraint Equations Used to Express Frictionless Slip of Multibody Systems Attached to Finite Elements—Application to a Contact between a Double Pendulum and a Beam
PublicationThis paper focuses on the numerical modeling of the dynamics of mechanical systems. Robots that can inspect high-voltage lines inspired this research. Their control systems must anticipate potential grab positions appropriately. We intend to formulate equations dedicated to the numerical description of the robot/cable contact. The investigated problem is not straightforward, since parts of the modeled systems are numerically inhomogeneous....
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A comprehensive approach to double inverted pendulum modelling
PublicationThe problem of mathematical modelling and indication of properties of a DIP has been investigated in this paper. The aim of this work is to aggregate the knowledge on a DIP modelling using the Euler-Lagrange formalism in the presence of external forces and friction. To indicate the main properties important for simulation, model parameters identification and control system synthesis, analytical and numerical tools have been used....
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Explicit and implicit difefrence methods for quasilinear first order partial functional differential equations.
PublicationInitial boundary value problems of the Dirichlet type for quasilinear functional differential equations are considered. Explicit difference schemes of the Euler type and implicit difference methods are investigated. Suffcient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented. It is proved that assumptions on the regularity of given functions are the same for both classes...
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Modal Adjustment of Rayleigh Based Structural Damping and Coordinate-Partitioning Algorithm Dedicated to Frictionless Contact Constraints between Multibody System and Structure Modelled with Finite Elements
PublicationThe paper presents a dedicated numerical algorithm. The algorithm is advantageous during investigations of the dynamics of a hybrid multibody / finite-elements system. We focus our attention on interactions resulting from mechanical contact. Pointwise contact connects a vertex of the multibody structure and a surface of the elastic reference body. Instead of a positive value of the relative penetration factor, constraint equations...
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General solution of quantum mechanical equations of motion with time-dependent Hamiltonians: A Lie algebraic approach
PublicationThe unitary operators U(t), describing the quantum time evolution of systems with a time-dependent Hamiltonian, can be constructed in an explicit manner using the method of time-dependent invariants. We clarify the role of Lie-algebraic techniques in this context and elaborate the theory for SU(2) and SU(1,1). In these cases we give explicit formulae for obtaining general solutions from special ones. We show that the constructions...
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Elastoplastic material law in 6-parameter nonlinear shell theory
PublicationWe develop the elastoplastic constitutive relations for nonlinear exact 6-parameter shell theory. A J2-type theory with strain hardening is formulated that takes into account asymmetric membrane strain measures. The incremental equations are solved using implicit Euler scheme with closest point projection algorithm. The presented test example shows the correctness of the proposed approach. Influence of micropolar material parameters...
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From fluid mechanics backgrounds to modern field theory
PublicationOur presentation keeps a historical line of reasoning, since we start from old concepts of fluid mechanics and finish on concepts of modern field theory. We want to show that some facts from the nature phenomena, which have firstly been discovered on the ground of fluid mechanics, were next incorporated into physics and later become the important pattern for whole mathematical physics. Especially, well-known continuum models, which...
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Nonlinear planar modeling of massive taut strings travelled by a force-driven point-mass
PublicationThe planar response of horizontal massive taut strings, travelled by a heavy point-mass, either driven by an assigned force, or moving with an assigned law, is studied. A kinematically exact model is derived for the free boundary problem via a variational approach, accounting for the singularity in the slope of the deflected string. Reactive forces exchanged between the point-mass and the string are taken into account via Lagrange...
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On forced vibrations of piezo-flexomagnetic nano-actuator beams
PublicationThe effect of excitation frequency on the piezomagnetic Euler-Bernoulli nanobeam taking the flexomagnetic material phenomenon into consideration is investigated in this chapter. The magnetization with strain gradients creates flexomagneticity. We couple simultaneously the piezomagnetic and flexomagnetic properties in an inverse magnetization. Resemble the flexoelectricity, the flexomagneticity is also size-dependent. So, it has...
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Vibration of the bridge under moving singular loads - theoretical formulation and numerical solution
PublicationThe paper presents the results of the numerical analysis of a simple vehicle passing over a simply supported bridge span. The bridge is modelled by a Euler-Bernoulli beam. The vehicle is modelled as a linear, visco-elastic oscillator, moving at a constant speed. The system is described by a set of differential equations of motion and solved numerically using the Runge-Kutta algorithm. The results are compared with the solution...