Discrete Euler-Lagrange Equations Research Articles

Difference Discrete Variational Principles, Euler–Lagrange Cohomology and Symplectic, Multisymplectic Structures I: Difference Discrete Variational Principle**The project supported by National Natural Science Foundation of China and National Key Project for Basic Research of China (G1998030601)

In this first paper of a series, we study the difference discrete variational principle in the framework of multi-parameter differential approach by regarding the forward difference as an entire geometric object in view of noncommutative differential geometry. Regarding the difference as an entire geometric object, the difference discrete version of Legendre transformation can be introduced. By virtue of this variational principle, we can discretely deal with the variation problems in both the Lagrangian and Hamiltonian formalisms to get difference discrete Euler–Lagrange equations and canonical ones for the difference discrete versions of the classical mechanics and classical field theory.

Discrete Euler-Poincaré and Lie-Poisson equations

In this paper, discrete analogues of Euler-Poincaré and Lie-Poisson reduction theory are developed for systems on finite dimensional Lie groups G with Lagrangians L:TG that are G-invariant. These discrete equations provide `reduced' numerical algorithms which manifestly preserve the symplectic structure. The manifold G × G is used as an approximation of TG, and a discrete Langragian :G × G is constructed in such a way that the G-invariance property is preserved. Reduction by G results in a new `variational' principle for the reduced Lagrangian :G, and provides the discrete Euler-Poincaré (DEP) equations. Reconstruction of these equations recovers the discrete Euler-Lagrange equations developed by Marsden et al (Marsden J E, Patrick G and Shkoller S 1998 Commun. Math. Phys. 199 351-395) and Wendlandt and Marsden (Wendlandt J M and Marsden J E 1997 Physica D 106 223-246) which are naturally symplectic-momentum algorithms. Furthermore, the solution of the DEP algorithm immediately leads to a discrete Lie-Poisson (DLP) algorithm. It is shown that when G = SO(n), the DEP and DLP algorithms for a particular choice of the discrete Lagrangian are equivalent to the Moser-Veselov scheme for the generalized rigid body.

A method for defining analog circuits for the minimization of discrete functionals: An image processing application

The solutions of many physical-mathematical problems can be obtained by minimizing proper functionals. In the literature, some methods for the synthesis of analog circuits (mainly cellular neural networks) are presented that find the solution of some of these problems by implementing the discretized Euler-Lagrange equations associated with the pertinent functionals. In this paper, we propose a method for defining analog circuits that directly minimize (in a parallel way) a class of discretized functionals in the frequently occurring case where the solution depends on two spatial variables. The method is a generalization of the one presented in Parodi et al.,Internat. J. Circuit Theory Appl., 26, 477---498, 1998. The analog circuits consist of both a (nonlinear) resistive part and a set of linear capacitors, whose steady-state voltages represent the discrete solution to the problem. The method is based on the potential (co-content) functions associated with voltage-controlled resistive elements. es an example, we describe an application in the field of image processing: the restoration of color images corrupted by additive noise.

Discrete Lagrangian Reduction, Discrete Euler–Poincaré Equations, and Semidirect Products

A discrete version of Lagrangian reduction is developed within the context of discrete time Lagrangian systems on G × G, where G is a Lie group. We consider the case when the Lagrange function is invariant with respect to the action of an isotropy subgroup of a fixed element in the representation space of G. Within this context, the reduction of the discrete Euler–Lagrange equations is shown to lead to the so-called discrete Euler–Poincare equations. A constrained variational principle is derived. The Legendre transformation of the discrete Euler–Poincare equations leads to discrete Hamiltonian (Lie–Poisson) systems on a dual space to a semiproduct Lie algebra.

Infant babbling in relation to ambient language

Grasping is a complex human activity performed with readiness through a complicated mechanical system as an end effector, i.e. the human hand. Here, we apply a direct transcription method of discrete mechanics and optimal control with constraints (DMOCC) to reproduce human-level grasping of an object with a three-dimensional model of the hand, actuated through joint control torques. The equations of motions describing the hand dynamics are derived from a discrete variational principle based on a discrete action functional, which gives the time integrator structure-preserving properties. The grasping action is achieved through a series of constraints, which generate a hybrid dynamical system with a given switching sequence and unknown switching times. To determine a favourable trajectory for grasping action, we solve an optimal control problem (OCP) with different physiological objectives subject to discrete Euler–Lagrange equations, boundary conditions and path constraints.

Methods for the Efficient Calculation of the (Mhd) Magnetohydrodynamic Stability Properties of Magnetically Confined Fusion Plasmas

A magnetohydrodynamic (MHD) model is applied to the problem of the stability of magnetically confined ther monuclear plasmas of interest in the pursuit of fusion power. Previous studies limited to two-dimensional con figurations are here generalized to three-dimensional toroidal plasmas. Using finite Fourier representations in the angle coordinates and finite hybrid elements in the radial direction, we solve the discretized Euler-Lagrange equations to determine the linear stability properties of the plasma.

Application of general discrete orthogonal polynomials to optimal-control systems

The shift-transformation matrix of general discrete orthogonal polynomials is introduced. General discrete orthogonal polynomials are adopted to obtain the modified discrete Euler-Lagrange equations. Then general discrete orthogonal polynomials are applied to simplify the discrete Euler-Lagrange equations into a set of linear algebraic ones for the approximation of state and control variables of digital systems. An example is included to demonstrate the simplicity and applicability of the method. Also, a comparison of the results obtained via several classical discrete orthogonal polynomials for the same problem is given.

Optimal control of digital systems via discrete Legendre polynomials

The shift transformation matrix for discrete Legendre polynomials is introduced in this study, and modified discrete Euler-Lagrange equations are constructed using the discrete variational principle combined with the idea of a penalty function. The discrete Legendre series are then applied to simplify the modified equations into a set of linear algebraic ones for the approximations of state and control variables of digital systems. An example is given to demonstrate this technique, when it is seen that this new method is straightforward and simple, giving a considerable saving in computing time.

Application of discrete Chebyshev polynomials to the optimal control of digital systems

The shift transformation matrix for discrete Chebyshev polynomials is introduced in this study. The discrete variational principle combined with the idea of penalty function is taken to construct the modified discrete Euler-Lagrange equations. Then, the discrete Chebyshev series are applied to simplify the modified equations into a set of linear algebraic ones for the approximations of state and control variables of digital systems. It is seen that this technique is quite straightforward and simple, and computing time can be saved considerably.

Optimal control using discrete Laguerre polynomials

This paper presents discrete Laguerre polynomials which approximate the optimal control problems of digital systems. The discrete variational principle, in combination with the idea of a penalty function, is used to construct the modified discrete Euler-Lagrange equations. The discrete Laguerre series is then applied to simplify the modified equations into a set of linear algebraic ones for the approximate solutions of state and control variables. An example is given, and the results obtained are very accurate and attractive.