Generalized Euler Lagrange Equations Research Articles

A uniform explanation of all falling chain phenomena

The many variants of falling chain problems that have been published in the physics literature over the last two centuries are here analyzed using generalized Euler–Lagrange equations that include momentum flux. Familiar results are recovered, and new ones are presented. We demonstrate and clarify the role played by momentum flux by considering the falling chains as both variable-mass and fixed-mass systems. We calculate the evolution of energy as a useful parameter that provides insight into some of the unexpected behavior of falling chain systems. Our Lagrangian approach incorporates all published falling chain phenomena (including the whiplash effect) within a unifying framework, highlighting connections between the phenomena and providing a valuable pedagogical example of variable mass systems analysis.

Variational problems of Herglotz type with complex order fractional derivatives and less regular Lagrangian

We derive optimality conditions for variational problems of Herglotz type whose Lagrangian depends on fractional derivatives of both real and complex order, and resolve the case of subdomain when the lower bounds of variational integral and fractional derivatives differ. Moreover, we consider a problem of the Herglotz type that corresponds to the case when the Lagrangian depends on the fractional derivative of the action and give an example of the problem that corresponds to the oscillator with a memory. Since our assumptions on the Lagrangian are weaker than in the classical theory, we analyze generalized Euler–Lagrange equations by the use of weak derivatives and the appropriate technics of distribution theory. Such an example is discussed in detail.

Variational and Optimal Control Approaches for the Second-Order Herglotz Problem on Spheres

The present paper extends the classical second-order variational problem of Herglotz type to the more general context of the Euclidean sphere S^n following variational and optimal control approaches. The relation between the Hamiltonian equations and the generalized Euler-Lagrange equations is established. This problem covers some classical variational problems posed on the Riemannian manifold S^n such as the problem of finding cubic polynomials on S^n. It also finds applicability on the dynamics of the simple pendulum in a resistive medium.

A Variational Principle for Discrete Integrable Systems

For integrable systems in the sense of multidimensional consistency (MDC) we can consider the Lagrangian as a form, which is closed on solutions of the equations of motion. For 2-dimensional systems, described by partial difference equations with two independent variables, MDC allows us to define an action on arbitrary 2-dimensional surfaces embedded in a higher dimensional space of independent variables, where the action is not only a functional of the field variables but also the choice of surface. It is then natural to propose that the system should be derived from a variational principle which includes not only variations with respect to the dependent variables, but also with respect to variations of the surface in the space of independent variables. Here we derive the resulting system of generalized Euler-Lagrange equations arising from that principle. We treat the case where the equations are 2 dimensional (but which due to MDC can be consistently embedded in higher-dimensional space), and show that they can be integrated to yield relations of quadrilateral type. We also derive the extended set of Euler-Lagrange equations for 3-dimensional systems, i.e., those for equations with 3 independent variables. The emerging point of view from this study is that the variational principle can be considered as the set of equations not only encoding the equations of motion but as the defining equations for the Lagrangians themselves.

Formulation of Euler–Lagrange Equations for Multidelay Fractional Optimal Control Problems

In this paper, a new numerical scheme is proposed for multidelay fractional order optimal control problems where its derivative is considered in the Grunwald–Letnikov sense. We develop generalized Euler–Lagrange equations that results from multidelay fractional optimal control problems (FOCP) with final terminal. These equations are created by using the calculus of variations and the formula for fractional integration by parts. The derived equations are then reduced into system of algebraic equations by using a Grunwald–Letnikov approximation for the fractional derivatives. Finally, for confirming the accuracy of the proposed approach, some illustrative numerical examples are solved.

Approximate Solutions of Nonsmooth Systems via Generalized Euler-Lagrange and Hamiltonian Equations

Approximate Solutions of Nonsmooth Systems via Generalized Euler-Lagrange and Hamiltonian Equations

Generalized Euler–Lagrange equations for fuzzy variational problems

We prove necessary optimality conditions for fuzzy variational problems by using the generalized Hukuhara differentiability concept. The fuzzy basic problem of the calculus of variations with free boundary conditions is considered, as well as problem with holonomic constraints. Examples are considered to demonstrate the applications of the new Euler–Lagrange equations.

A new Bernoulli–Euler beam model based on a simplified strain gradient elasticity theory and its applications

A new Bernoulli–Euler beam model based on a simplified strain gradient elasticity theory is established in the current investigation. The generalized Euler–Lagrange equations and corresponding boundary conditions are naturally derived from the Hamilton’s principle. Then axial wave propagation of small scale bars, static bending of cantilever beams, buckling and free vibration of simply supported beams are analytically solved by using the simplified strain gradient beam theory. The influences of the Poisson’s effect as well as the weak non-local strain gradient elastic effect are discussed. The Poisson’s effect is found to increase with the increase of the beam thickness in the buckling analysis, while the higher-order bending moment induced by stretch strain gradient has an insignificant influence on the critical buckling load in our numerical analysis. However, the effect of the higher-order bending moment is very significant on axial wave propagation and static bending of micro-scale beams. The current work is very helpful in understanding the microstructure-related size dependent phenomenon.

Global geometry of non-planar 3-body motions

The aim of this paper is to study the global geometry of non-planar 3-body motions in the realms of equivariant Differential Geometry and Geometric Mechanics. This work was intended as an attempt at bringing together these two areas, in which geometric methods play the major role, in the study of the 3-body problem. It is shown that the Euler equations of a three-body system with non-planar motion introduce non-holonomic constraints into the Lagrangian formulation of mechanics. Applying the method of undetermined Lagrange multipliers to study the dynamics of three-body motions reduced to the level of moduli space \({\bar{M}}\) subject to the non-holonomic constraints yields the generalized Euler-Lagrange equations of non-planar three-body motions in \({\bar{M}}\) . As an application of the derived dynamical equations in the level of \({\bar{M}}\) , we completely settle the question posed by A. Wintner in his book [The analytical foundations of Celestial Mechanics, Sections 394–396, 435 and 436. Princeton University Press (1941)] on classifying the constant inclination solutions of the three-body problem.

Generalized Euler–Lagrange equations for fractional variational problems with free boundary conditions

In this article we are going to present necessary conditions which must be satisfied to make the fractional variational problems (FVPs) with completely free boundary conditions have an extremum. The fractional derivatives are defined in the Caputo sense. First we present the necessary conditions for the problem with only one dependent variable, and then we generalize them to problems with multiple dependent variables. We also find the transversality conditions for when each end point lies on a given arbitrary curve in the case of a single variable or a surface in the case of multiple variables. It is also shown that in special cases such as those with specified and unspecified boundary conditions and problems with integer order derivatives, the new results reduce to the known necessary conditions. Some examples are presented to demonstrate the applicability of the new formulations.

Generalized Euler–Lagrange Equations for Variational Problems with Scale Derivatives

We obtain several Euler-Lagrange equations for variational functionals defined on a set of H\"older curves. The cases when the Lagrangian contains multiple scale derivatives, depends on a parameter, or contains higher-order scale derivatives are considered.

Generalized Euler—Lagrange Equations and Transversality Conditions for FVPs in terms of the Caputo Derivative

This paper presents extensions to the traditional calculus of variations for systems containing Fractional Derivatives (FDs) defined in the Caputo sense. Specifically, two problems are considered, the simplest Fractional Variational Problem (FVP) and the FVP of Lagrange. Both specified and unspecified end conditions and end points are considered. Results of the first problem are extended to problems containing multiple FDs. For the second problem, we present a Lagrange type multiplier rule. For both problems, we develop generalized Euler—Lagrange type necessary conditions and the transversality conditions, which must be satisfied for the given functional to be extremum. The formulations show that one may require fractional boundary conditions even when the FVP has been formulated in terms of Caputo Fractional Derivatives (CFDs) only, and that the Riemann—Liouville Fractional Derivatives (RLFDs) automatically appear in the resulting equations. The relationships between the RLFDs and the CFDs are used to write the resulting differential equations purely in terms of the CFDs. Several examples are considered to demonstrate the application of the formulations. The formulations presented and the resulting equations are very similar to the formulations given for FVPs defined in terms of the RLFDs and to those that appear in the field of classical calculus of variations. The formulations presented are simple and can be extended to other problems in the field of fractional calculus of variations.

Fractional variational calculus in terms of Riesz fractional derivatives

This paper presents extensions of traditional calculus of variations for systems containing Riesz fractional derivatives (RFDs). Specifically, we present generalized Euler–Lagrange equations and the transversality conditions for fractional variational problems (FVPs) defined in terms of RFDs. We consider two problems, a simple FVP and an FVP of Lagrange. Results of the first problem are extended to problems containing multiple fractional derivatives, functions and parameters, and to unspecified boundary conditions. For the second problem, we present Lagrange-type multiplier rules. For both problems, we develop the Euler–Lagrange-type necessary conditions which must be satisfied for the given functional to be extremum. Problems are considered to demonstrate applications of the formulations. Explicitly, we introduce fractional momenta, fractional Hamiltonian, fractional Hamilton equations of motion, fractional field theory and fractional optimal control. The formulations presented and the resulting equations are similar to the formulations for FVPs given in Agrawal (2002 J. Math. Anal. Appl. 272 368, 2006 J. Phys. A: Math. Gen. 39 10375) and to those that appear in the field of classical calculus of variations. These formulations are simple and can be extended to other problems in the field of fractional calculus of variations.

Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem

This paper extends the generalized variational principle of Herglotz to one with several independent variables and derives the corresponding generalized Euler–Lagrange equations. The extended principle contains the classical variational principle with several independent variables and the variational principle of Herglotz as special cases. A first Noether-type theorem is proven for the new variational principle, which gives the conserved quantities corresponding to symmetries of the associated functional. This theorem contains the classical first Noether theorem as a special case. As examples for applications we calculate a conserved quantity for the damped nonlinear Klein–Gordon equation and we show that the equations which describe the propagation of electromagnetic fields in a conductive medium can be derived from the generalized variational principle of Herglotz (but not from a classical variational principle).

The generalized Lagrange formulation for nonlinear RLC networks

Based on the concept of generalized Euler-Lagrange equations, this paper develops a Lagrange formulation of RLC networks of considerably broad scope. It is shown that the generalized Lagrange equations along with a set of compatibility constraint equations represents a set of governing differential equations of order equal to the order of complexity of the network. In this method the generalized coordinates include capacitor charges and inductor fluxes and the generalized velocities are comprised of an independent set of capacitor voltages and inductor currents. The generalized Hamilton equations are also developed and the connection with the Brayton-Moser equations is established.