Variational sequences in mechanics

Let be a fibered manifold over a base manifold . A differential 1-form , defined on the -jet prolongation of , is said to be contact, if it vanishes along the -jet prolongation of every section of . The notion of contactness is naturally extended to -forms with . The contact forms define a subsequence of the De Rham sequence on . The corresponding quotient sequence is known as the rth order variational sequence. In this paper, the case of 1-dimensional base is considered. A simple proof is given of the fact that the rth order variational sequence is an acyclic resolution of the constant sheaf. Then the 1st order variational sequence is studied in detail. The quotient sheaves, as well as the quotient mappings, are determined explicitly, and their relationship to the standard concepts of the 1st order calculus of variations is discussed. The following is shown: a) the lagrangians in the 1st order variational sequence (classes of 1-forms) coincide with 2nd order lagrangians, affine in the second derivative variables, b) the concept of the Euler-Lagrange form is extended to 2-forms which are not necessarily variational, c) the concept of the Helmholtz-Sonin form is introduced as the class of an arbitrary 3-form, d) the well-known fundamental notions such as the Euler-Lagrange, and Helmholtz-Sonin mappings are represented by two arrows at the beginning of the variational sequence; this relates the global structure of the Euler-Lagrange mapping to the cohomology of , e) all the remaining classes of -forms with , as well as the quotient mappings, are determined explicitly, f) a locally variational form is defined as a generalization of a symplectic form; locally variational forms, associated to a fixed Euler-Lagrange form, are characterized, and g) distributions associated with a locally variational form are described, and their relation to the Euler-Lagrange equations is studied. These results illustrate differences between finite order variational sequences and variational bicomplexes, based on infinite jet constructions. Received February 18, 1996 / In revised form December 1996 / Accepted December 2, 1996     

Variational Sequences in Mechanics on Grassmann Fibrations

Extension of the variational sequence theory in mechanics to the first order Grassmann fibrations of 1-dimensional submanifolds is presented. The correspondence with the variational theory of parameter-invariant problems on manifolds is discussed in terms of the theory of jets (slit tangent bundles) and contact elements. In particular, the Helmholtz expressions for parameter-invariant variational problems, measuring local variationality of differential forms and differential equations, are given in the canonical and adapted coordinates. The methods can easily be extended to higher order variational problems.     

Contact symmetries of the Helmholtz form

In this paper, vector fields which are symmetries of the contact ideal are studied. It is shown that contact symmetries of the Helmholtz form transform a dynamical form to a dynamical form which is variational (i.e. comes as the Euler-Lagrange form from a Lagrangian). The case of dynamical forms representing first-order classes in the variational sequence is analysed in detail, which means, by the variational sequence theory, that systems of ordinary differential equations of order ?3 are concerned.     

Forms of Lepage type and the balance systems

In this work we apply infinitesimal variational calculus to the systems of balance equations. We determine a class of the exterior $n+(n+1)$-forms Θ on the jet bundle of infinite order over a configurational bundle $\pi :Y^{n+m}\to X^n$ similar to the class of Lepage n-forms. Systems of differential equations obtained in the way similar to one used in the Lagrangian field theory, include the Euler–Lagrange equations corresponding to a Lagrangian functions as well as arbitrary regular systems of balance equations. For a balance system with a symmetry group G we present the Noether balance laws corresponding to the generators of the Lie algebra of the group G.     

Input–output control systems and dichotomy of variational difference equations

We propose a new and unified approach for the study of dichotomy of variational difference equations, establishing a link between control methods and basic techniques from interpolation theory. We obtain necessary and sufficient conditions for the existence of uniform dichotomy and, respectively, for uniform exponential dichotomy of variational difference equations in terms of the admissibility of general pairs of sequence spaces. We provide a classification of the main classes of sequence spaces where the input spaces and the output spaces may belong to, for each dichotomy property and prove that the hypotheses on the underlying sequence spaces cannot be removed. The obtained results extend the framework to the study of dichotomy of variational difference equations, hold without any requirement on the coefficients and are applicable to all systems of variational difference equations.     

A new approach to nonlinear partial differential equations

In this paper, a novel method called variational iteration method is proposed to solve nonlinear partial differential equations without linearization or small perturbations. In this method, a correction functional is constructed by a general Lagrange multiplier, which can be identified via variational theory. An analytical solution can be obtained from its trial-function with possible unknown constants, which can be identified by imposing the boundary conditions, by successively iteration.     

Hamiltonian and Variational Linear Distributed Systems

We use the formalism of bilinear- and quadratic differential forms in order to study Hamiltonian and variational linear distributed systems. It was shown in [1] that a system described by ordinary linear constant-coefficient differential equations is Hamiltonian if and only if it is variational. In this paper we extend this result to systems described by linear, constant-coefficient partial differential equations. It is shown that any variational system is Hamiltonian, and that any scalar Hamiltonian system is contained (in general, properly) in a particular variational system.     

Variational iteration method for solving non-linear partial differential equations

In this paper, we shall use the variational iteration method to solve some problems of non-linear partial differential equations (PDEs) such as the combined KdV–MKdV equation and Camassa–Holm equation. The variational iteration method is superior than the other non-linear methods, such as the perturbation methods where this method does not depend on small parameters, such that it can fined wide application in non-linear problems without linearization or small perturbation. In this method, the problems are initially approximated with possible unknowns, then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory.     

Non-integrability of Hamiltonian systems through high order variational equations: Summary of results and examples

This paper deals with non-integrability criteria, based on differential Galois theory and requiring the use of higher order variational equations. A general methodology is presented to deal with these problems. We display a family of Hamiltonian systems which require the use of order k variational equations, for arbitrary values of k, to prove non-integrability. Moreover, using third order variational equations we prove the non-integrability of a non-linear spring-pendulum problem for the values of the parameter that can not be decided using first order variational equations.      

Riemannian convexity of functionals

This paper extends the Riemannian convexity concept to action functionals defined by multiple integrals associated to Lagrangian differential forms on first order jet bundles. The main results of this paper are based on the geodesic deformations theory and their impact on functionals in Riemannian setting. They include the basic properties of Riemannian convex functionals, the Riemannian convexity of functionals associated to differential m-forms or to Lagrangians of class C ¹ respectively C ², the generalization to invexity and geometric meaningful convex functionals. Riemannian convexity of functionals is the central ingredient for global optimization. We illustrate the novel features of this theory, as well as its versatility, by introducing new definitions, theorems and algorithms that bear upon the currently active subject of functionals in variational calculus and optimal control. In fact so deep rooted is the convexity notion that nonconvex problems are tackled by devising appropriate convex approximations.     

Nonintegrability of Parametrically Forced Nonlinear Oscillators

We discuss nonintegrability of parametrically forced nonlinear oscillators which are represented by second-order homogeneous differential equations with trigonometric coefficients and contain the Duffing and van der Pol oscillators as special cases. Specifically, we give sufficient conditions for their rational nonintegrability in the meaning of Bogoyavlenskij, using the Kovacic algorithm as well as an extension of the Morales–Ramis theory due to Ayoul and Zung. In application of the extended Morales–Ramis theory, for the associated variational equations, the identity components of their differential Galois groups are shown to be not commutative even if the differential Galois groups are triangularizable, i. e., they can be solved by quadratures. The obtained results are very general and reveal their rational nonintegrability for the wide class of parametrically forced nonlinear oscillators. We also give two examples for the van der Pol and Duffing oscillators to demonstrate our results.     

Spencer sequence and variational sequence

The purpose of this paper is to revisit the construction of the variational sequence existing within the formal calculus of variations, in order to stabilize the order of jets involved and to establish a link with the dual of the Spencer sequence existing within the formal theory of systems of partial differential equations.     

Relations between symmetries and conservation laws for difference systems

This paper presents a relation between divergence variational symmetries for difference variational problems on lattices and conservation laws for the associated Euler–Lagrange system provided by Noether's theorem. This inspires us to define conservation laws related to symmetries for arbitrary difference equations with or without Lagrangian formulations. These conservation laws are constrained by partial differential equations obtained from the symmetries generators. It is shown that the orders of these partial differential equations have been reduced relative to those used in a general approach. Illustrative examples are presented.     

New efficient numerical procedures for solving stochastic variational problems with a priori maximum pointwise error estimates

In a previous paper we gave a new formulation and derived the Euler equations and other necessary conditions to solve strong, pathwise, stochastic variational problems with trajectories driven by Brownian motion. Thus, unlike current methods which minimize the control over deterministic functionals (the expected value), we find the control which gives the critical point solution of random functionals of a Brownian path and then, if we choose, find the expected value.This increase in information is balanced by the fact that our methods are anticipative while current methods are not. However, our methods are more directly connected to the theory and meaningful examples of deterministic variational theory and provide better means of solution for free and constrained problems. In addition, examples indicate that there are methods to obtain nonanticipative solutions from our equations although the anticipative optimal cost function has smaller expected value.In this paper we give new, efficient numerical methods to find the solution of these problems in the quadratic case. Of interest is that our numerical solution has a maximal, a priori, pointwise error of O(h3/2) where h is the node size. We believe our results are unique for any theory of stochastic control and that our methods of proof involve new and sophisticated ideas for strong solutions which extend previous deterministic results by the first author where the error was O(h²).We note that, although our solutions are given in terms of stochastic differential equations, we are not using the now standard numerical methods for stochastic differential equations. Instead we find an approximation to the critical point solution of the variational problem using relations derived from setting to zero the directional derivative of the cost functional in the direction of simple test functions.Our results are even more significant than they first appear because we can reformulate stochastic control problems or constrained calculus of variations problems in the unconstrained, stochastic calculus of variations formulation of this paper. This will allow us to find efficient and accurate numerical solutions for general constrained, stochastic optimization problems. This is not yet being done, even in the deterministic case, except by the first author.     

Higher order conservation laws and a higher order Noether's theorem

Consider a general variational problem of a functional whose domain of definition consists of integral manifolds of an exterior differential system. In particular, this induces classical variational problems with constraints. With the assumption of existence of enough admissable variations the Euler-Lagrange equations associated to this problem are obtained. By studying a spectral sequence associated to the infinite prolongation of them, we extend the classical notion of infinitesimal Noether symmetries to what we shall call the “higher order Noether symmetries,” and a higher order Noether's theorem identifying the higher order conservation laws and the higher order Noether symmetries is obtained. These in turn are isomorphic to the solution space of certain linear differential operator. From these we also get a systematic method of computing the higher order conservation laws of certain determined PDE systems.     

Variational iteration method for solving the space‐ and time‐fractional KdV equation

This paper presents numerical solutions for the space‐ and time‐fractional Korteweg–de Vries equation (KdV for short) using the variational iteration method. The space‐ and time‐fractional derivatives are described in the Caputo sense. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers in the functionals can be identified optimally via variational theory. The iteration method, which produces the solutions in terms of convergent series with easily computable components, requiring no linearization or small perturbation. The numerical results show that the approach is easy to implement and accurate when applied to space‐ and time‐fractional KdV equations. The method introduces a promising tool for solving many space–time fractional partial differential equations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

Elliptic partial differential equation and optimal control

The theory of optimal control and the semianalytical method of elliptic partial differential equation (PDE) in a prismatic domain are mutually simulated issues. The simulation of discrete-time linear quadratic (LQ) control with the substructural chain problem in static structural analysis is given first. From the minimum potential energy variational principle of substructural chain, the generalized variational principle with two kinds of variables and the dual equations are derived. The simulation relation is then recognized by comparing the variational principle and dual equations of the LQ control theory. The simulation between elliptic PDE in the prismatic domain and continuous-time LQ control is established in the same way, and the interval energy is naturally introduced, as in the case of substructural chain. The assembling and condensation equations can help one to derive the differential equations of the submatrices of potential energy and mixed energy. The well known Riccati equation is one of them. The interval assembling and condensation algorithm can be used to solve the Riccati equation. Some numerical examples are given to illustrate the method.     

<Emphasis Type="Italic">r</Emphasis>-Tuple almost product structures

A generalization of an almost product structure and an almost complex structure on smooth manifolds is constructed. The set of tensor differential invariants of type (2, 1) and the set of differential 2-forms for such structures are constructed. We show how these tensor invariants can be used to solve the classification problem for Monge–Ampère equations and Jacobi equations.     

Finite-Order Formulation of Vinogradov's C-Spectral Sequence

The C-spectral sequence was introduced by A. M. Vinogradov in the late Seventies as a fundamental tool for the study of the algebro-geometric properties of jet spaces and differential equations. A spectral sequence arises from the contact filtration of the modules of forms on jet spaces of a fibring (or on a differential equation). In order to avoid serious technical difficulties, the order of the jet space is not fixed, i.e., computations are performed on spaces containing forms on jet spaces of any order. In this paper we show that there exists a formulation of Vinogradov's C-spectral sequence in the case of finite-order jet spaces of a fibred manifold. We compute all cohomology groups of the finite-order C-spectral sequence. We obtain a finite-order variational sequence which is shown to be naturally isomorphic with Krupka's finite-order variational sequence.