Order Euler-Lagrange Equations Research Articles

Generalized N-rotor problems, synchronized subsystems, and associated solitons.

We consider systems of N particles interacting on the unit circle through 2π-periodic potentials. An example is the N-rotor problem that arises as the classical limit of coupled Josephson junctions and for various energies is known to have a wide range of behaviors such as global chaos and ergodicity, together with families of periodic solutions and transitions from order to chaos. We focus here on selected initial values for generalized systems in which the second order Euler-Lagrange equations reduce to first order equations, which we show by example can describe an ensemble of oscillators with associated emergent phenomena such as synchronization. A specific case is that of the Kuramoto model with well-known synchronization properties. We further demonstrate the relation of these models to field theories in 1+1 dimensions that allow static kink solitons satisfying first order Bogomolny equations, well-known in soliton physics, which correspond to the first order equations of the generalized N-rotor models. For the nonlinear pendulum, for example, the first order equations define the separatrix in the phase portrait of the system and correspond to kink solitons in the sine-Gordon equation.

Calculus of variations and optimal control for generalized functions

We present an extension of some results of higher order calculus of variations and optimal control to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove the higher order Euler–Lagrange equations, the D’Alembert principle in differential form, the du Bois–Reymond optimality condition and the Noether’s theorem. We start the theory of optimal control proving a weak form of the Pontryagin maximum principle and the Noether’s theorem for optimal control. We close with a study of a singularly variable length pendulum, oscillations damped by two media and the Pais–Uhlenbeck oscillator with singular frequencies.

Second order Lagrangian dynamics on double cross product groups

We observe that the iterated tangent group of a Lie group may be realized as a double cross product of the 2nd order tangent group, with the Lie algebra of the base Lie group. Based on this observation, we derive the 2nd order Euler–Lagrange equations on the 2nd order tangent group from the 1st order Euler–Lagrange equations on the iterated tangent group. We also present in detail the 2nd order Lagrangian dynamics on the 2nd order tangent group of a double cross product group.

Higher order mechanics on graded bundles

In this paper we develop a geometric approach to higher order mechanics on graded bundles in both, the Lagrangian and Hamiltonian formalism, via the recently discovered weighted algebroids. We present the corresponding Tulczyjew triple for this higher order situation and derive in this framework the phase equations from an arbitrary (also singular) Lagrangian or Hamiltonian, as well as the Euler–Lagrange equations. As important examples, we geometrically derive the classical higher order Euler–Lagrange equations and analogous reduced equations for invariant higher order Lagrangians on Lie groupoids.

Fractional Euler-Lagrange Equations Applied to Oscillatory Systems

In this paper, we applied the Riemann-Liouville approach and the fractional Euler-Lagrange equations in order to obtain the fractional nonlinear dynamic equations involving two classical physical applications: “Simple Pendulum” and the “Spring-Mass-Damper System” to both integer order calculus (IOC) and fractional order calculus (FOC) approaches. The numerical simulations were conducted and the time histories and pseudo-phase portraits presented. Both systems, the one that already had a damping behavior (Spring-Mass-Damper) and the system that did not present any sort of damping behavior (Simple Pendulum), showed signs indicating a possible better capacity of attenuation of their respective oscillation amplitudes. This implication could mean that if the selection of the order of the derivative is conveniently made, systems that need greater intensities of damping or vibrating absorbers may benefit from using fractional order in dynamics and possibly in control of the aforementioned systems. Thereafter, we believe that the results described in this paper may offer greater insights into the complex behavior of these systems, and thus instigate more research efforts in this direction.

Coupled-Nonlinear Elastic Structure: An Innovative Parameterization Scheme of the Motion Equations

In this paper, I applied the Euler-Lagrange equations in order to obtain the coupled-nonlinear motion equations for an elastic structure. The model is composed of six coupled and strongly nonlinear ordinary differential equations. The new contribution of this work arises from the fact that a convenient and innovative parameterization of the motion equations for the elastic system was developed with all mathematical nonlinearities taken into account, without the usage of any simplifying linearization procedure, as found in most of the works presented in the literature. The results can be used as a source for conducting experiments and can be useful for a better understanding and control of such nonlinear elastic systems.

A Skyrme-like model with an exact BPS bound

We propose a new Skyrme-like model with fields taking values on the sphere S^3 or, equivalently, on the group SU(2). The action of the model contains a quadratic kinetic term plus a quartic term which is the same as that of the Skyrme-Faddeev model. The novelty of the model is that it possess a first order Bogomolny type equation whose solutions automatically satisfy the second order Euler-Lagrange equations. It also possesses a lower bound on the static energy which is saturated by the Bogomolny solutions. Such Bogomolny equation is equivalent to the so-called force free equation used in plasma and solar Physics, and which possesses large classes of solutions. An old result due to Chandrasekhar prevents the existence of finite energy solutions for the force free equation on the entire tridimensional space R^3. We construct new exact finite energy solutions to the Bogomolny equations for the case where the space is the three-sphere S^3, using toroidal like coordinates.

Augmented Lagrangian method for a mean curvature based image denoising model

High order derivative information has been widely used in developing variational models in image processing to accomplish more advanced tasks. However, it is a nontrivial issue to construct efficient numerical algorithms to deal with the minimization of these variational models due to the associated high order Euler-Lagrange equations. In this paper, we propose an efficient numerical method for a mean curvature based image denoising model using the augmented Lagrangian method. A special technique is introduced to handle the mean curvature model for the augmented Lagrangian scheme. We detail the procedures of finding the related saddle-points of the functional. We present numerical experiments to illustrate the effectiveness and efficiency of the proposed numerical method, and show a few important features of the image denoising model such as keeping corners and image contrast. Moreover, a comparison with the gradient descent method further demonstrates the efficiency of the proposed augmented Lagrangian method.

Fractional-order Euler–Lagrange equations and formulation of Hamiltonian equations

This paper presents the fractional order Euler–Lagrange equations and the transversality conditions for fractional variational problems with fractional integral and fractional derivatives defined in the sense of Caputo and Riemann–Liouville. A fractional Hamiltonian formulation was developed and some illustrative examples were treated in detail.

MOVING HORIZON SIMULTANEOUS ESTIMATION OF PROCESS GAIN AND DISTURBANCES FOR AN OXYGEN CONVERTER GAS RECOVERY PROCESS MODEL

This paper proposes a moving horizon simultaneous estimation of process gain and disturbances for discrete-time linear systems with unknown process gain so as to minimize a moving-horizon performance index. The proposed method uses discrete-time Euler-Lagrange equations in order to derive the proposed adaptive disturbance estimator. A numerical simulation for an oxygen converter gas recovery process shows the efficiency of the proposed method.

A new algorithm for solving differential/algebraic equations of multibody system dynamics

The second order Euler-Lagrange equations are transformed to a set of first order differentiallalgebraic equations, which are then transformed to state equations by using local parameierization. The corresponding discretization method is presented, and the results can be used to implementation of various numerical integration methods. A numerical example is presented finally.

The neutrino energy-momentum tensor and the Weyl equations in curved space-time

In general, a first order Lagrangian gives rise to second order Euler-Lagrange equations. However, there are important examples where the associated Euler-Lagrange equations are of first order only, the Weyl neutrino equations being of this type. In this paper we therefore consider first order spinor Lagrangians which give rise to firstorder Euler-Lagrange equations. Specifically, the most general first order spinor field equations of rank one in curved space-time which are derivable from a first order Lagrangian of the same type are explicitly constructed. Subject to a certain restriction, the Weyl neutrino equation is the only possibility. Furthermore, if the spinor field satisfies the Weyl neutrino equation, then the associated energy momentum tensor is the conventional neutrino energymomentum tensor.

Degenerate Lagrange densities involving geometric objects

The Euler-Lagrange equations corresponding to a Lagrange density which is a function of a symmetric affine connection, Γ i j h , and its first derivatives together with a symmetric tensor gi j, are investigated. In general, by variation of the Γ i j h , these equations will be of second order in Γ i j h . Necessary and sufficient conditions for these Euler-Lagrange equations to be of order one and zero in Γ i j h are obtained. It is shown that if the gi j may be regarded as independent then the only permissible zero order Euler-Lagrange equations are those which ensure that the Γ i j h are precisely the Christoffel symbols of the second kind.

The uniqueness of the Einstein field equations in a four-dimensional space

The Euler-Lagrange equations corresponding to a Lagrange density which is a function of g ij and its first two derivatives are investigated. In general these equations will be of fourth order in g ij. Necessary and sufficient conditions for these Euler-Lagrange equations to be of second order are obtained and it is shown that in a four-dimensional space the Einstein field equations (with cosmological term) are the only permissible second order Euler-Lagrange equations. This result is false in a space of higher dimension. Furthermore, the only permissible third order equation in the four-dimensional case is exhibited.