Formal Variational Calculus Research Articles

On the realization of transitive Novikov algebras

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in the formal variational calculus. It is well known that the radical of a finite-dimensional Novikov algebra is transitive. In this paper, we prove that a kind realization of Novikov algebras given by S Gel'fand is transitive and we give a deformation theory of Novikov algebras. In two and three dimensions, we find that all transitive Novikov algebras can be realized as the Novikov algebras given by S Gel'fand and their compatible infinitesimal deformations.

The classification of Novikov algebras in low dimensions

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in the formal variational calculus. For further our understanding and physical applications, we give a classification of Novikov algebras in dimensions two and three in this paper.

Transitive Novikov Algebras on Four-Dimensional Nilpotent Lie Algebras

Novikov algebras were introduced in connection with the Poisson brackets (of hydrodynamic type) and Hamiltonian operators in the formal variational calculus. The commutator of a Novikov algebra is a Lie algebra, and the radical of a finite-dimensional Novikov algebra is transitive. In this paper, we give a classification of transitive Novikov algebras on four-dimensional nilpotent Lie algebras based on Kim (1986, Journal of Differential Geometry 24, 373–394).

Bering’s proposal for boundary contribution to the Poisson bracket

It is shown that the Poisson bracket with boundary terms proposed by Bering can be deduced from the Poisson bracket proposed by the present author if one omits terms free of Euler–Lagrange derivatives (“annihilation principle”). This corresponds to another definition of the formal product of distributions (or, saying it in other words, to another definition of the pairing between 1-forms and 1-vectors in the formal variational calculus). We extend the formula (initially suggested by Bering for the ultralocal case with constant coefficients only) onto the general nonultralocal brackets with coefficients depending on fields and their spatial derivatives. The lack of invariance under changes of dependent variables (field redefinitions) seems to be a drawback of this proposal.

Variational Calculus of Supervariables and Related Algebraic Structures

We establish a formal variational calculus of supervariables, which is a combination of the bosonic theory of Gel'fand–Dikii and the fermionic theory in our earlier work. Certain interesting new algebraic structures are found in connection with Hamiltonian superoperators in terms of our theory. In particular, we find connections between Hamiltonian superoperators and Novikov–Poisson algebras that we introduced in our earlier work in order to establish a tensor theory of Novikov algebras. Furthermore, we prove that an odd linear Hamiltonian superoperator in our variational calculus induces a Lie superalgebra, which is a natural generalization of the super-Virasoro algebra under certain conditions.

Divergences in formal variational calculus and boundary terms in Hamiltonian formalism

It is shown how to extend the formal variational calculus in order to incorporate integrals of divergences into it. Such a generalization permits to study nontrivial boundary problems in field theory on the base of canonical formalism.

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.

Nonassociative algebras related to Hamiltonian operators in the formal calculus of variations

The main result in this paper is the classification of simple Novikov algebras A with a maximal subalgebra H such that A H has a finite-dimensional irreducible H-submodule. A second result deals with the extension of Hamiltonian operators.

Hamiltonian superoperators

We present a theory of Hamiltonian superoperators associated with a Lie superalgebra and its modules. By using the free fermionic fields from physics, we establish a super version of the formal variational calculus introduced by Gel'fand and Dikii (1975, 1976). Moreover, we prove that a super skew-symmetric matrix differential operator in our super formal variational calculus is a Hamiltonian operator if and only if its Schouten-Nijenhuis super-bracket is zero, when the characteristic of the base field is not two. Some interesting examples of Hamiltonian superoperators are also given.

Super Hamiltonian operators and Lie superalgebras

A superversion of the formal calculus of variations as developed by I.M. Gel'fand et al. is presented. This superversion is essentially contained in the one given by B.A. Kupershmidt. However, the form presented here is more manageable and suitable for applications. Even as well as odd super Hamiltonian operators are considered. It is proved that with every linear super Hamiltonian operator one can associate a Lie superalgebra structure on the space of (reduced) 1-forms and vice versa. Also a theorem is proved about the connection between 2-cocycles on this Lie superalgebra and super Hamiltonian operators. Finally an application to the sKdV equation is given.

Hamiltonian operators in graded formal calculus of variations

Hamiltonian operators in graded formal calculus of variations

Determinantal ideals and the Capelli identities

Recent work in the formal calculus of variations and also in Riemannian geometry and general relativity have relied upon certain important, previously established properties of determinant ideals in characteristic zero. This note presents direct, elementary proofs of these properties by utilizing the Capelli identities of classical invariant theory.

On formal variational calculus of higher dimensions

Based upon the definition of variational derivatives presented by Galindo and Martinez Alonso, a series of formulae on formal variational calculus of higher dimensions is developed.

Conservation laws generated by pairs of non-Hamiltonian symmetries

We consider systems of partial differential equations of the form w-here u = (u’,..., u”) are dependent variables and K is .a q-tuple of “partial differential functions,“- i.e., smooth functions of x = (or ,..., x,) and, the various derivatives- z& = 8ui, .I =,(jr ,..., j,), 8 = (3, ,“., aP), ai = 3/L+. The system is autonomous: K does not depend on the time t explicitly. Our point of view is that of the formal variational calculus as in Gel’fand and Dikii [I--4], Gel’fand and Dorfman [5.], and Olver [7-IO]. The essential elements of this calculus will be set out in Section 2. A scalar partial differential function T (which may also depend explicitly on time) is conserved under the flow of u, = K if i T dx is independent of time, ” IPP This is formally equivalent to D, T = a divergence expression, say, D,P, + -.- + D,D,, where the (Pi) are partial functions and {Di and D, are total derivatives. For example, whereas D,(tu’) = 2tuK + uJ,

Euler operators and conservation laws of the BBM equation

AbstractThe BBM or Regularized Long Wave Equation is shown to possess only three non-trivial independent conservation laws. In order to prove this result, a new theory of Euler-type operators in the formal calculus of variations will be developed in detail.

Formal variational calculus in the algebra of smooth cylindrical functions

Formal variational calculus in the algebra of smooth cylindrical functions

Poisson brackets and the kernel of the variational derivative in the formal calculus of variations

Poisson brackets and the kernel of the variational derivative in the formal calculus of variations

Stable and unstable elastica equilibrium and the problem of minimum curvature

The Euler-Bernoulli equations governing a stable equilibrium configuration of an elastica, subject to pin supports, can be derived by a formal calculus of variations via a local minimum principle for the strain energy of the elastica. Such a derivation was carried out by Lee and Forsythe [4] in their analysis of open and closed nonlinear spline curves. The basic equation satisfied for each unloaded span between supports by the curvature (or normalized bending moment) k is the familiar equation in the arc length variable

Invariant variation problems

The problems in variation here concerned are such as to admit a continuous group (in Lie's sense); the conclusions that emerge from the corresponding differential equations find their most general expression in the theorems formulated in Section 1 and proved in following sections. Concerning these differential equations that arise from problems of variation, far more precise statements can be made than about arbitrary differential equations admitting of a group, which are the subject of Lie's researches. What is to follow, therefore, represents a combination of the methods of the formal calculus of variations with those of Lie's group theory. For special groups and problems in variation, this combination of methods is not new; I may cite Hamel and Herglotz for special finite groups, Lorentz and his pupils (for instance Fokker), Weyl and Klein for special infinite groups. Especially Klein's second Note and the present developments have been mutually influenced by each other, in which regard I may refer to the concluding remarks of Klein's Note.