On the existence of variational principles for differential-difference evolution equations

Filippov, V. M.; Savchin, V. M.; Budochkina, S. A.

doi:10.1134/s0081543813080038

Cited by 4 publications

(3 citation statements)

References 3 publications

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“…These issues are closely related to the inverse problem of the calculus of variations (IPCV) in the following statement: for a given equation, one needs to construct a functional such that its set of stationary points coincides with the set of solutions to this equation. There is a large number of works devoted to inverse problems of the calculus of variations: for ordinary differential equations and partial differential equations [3], [4], [7], [9], [19], [20], [26], [27], operator equations [6], [21], [22], differential-difference equations [8], [17], [18], stochastic differential equations [23], [24], [25], fractional differential equations [1], [10], [14], [28]. In these works, nonlocal bilinear forms were mainly used to solve the IPCV.…”

Section: Introductionmentioning

confidence: 99%

On indirect representability of fourth order ordinary differential equation in form of Hamilton - Ostrogradsky equations

Budochkina,

Luu,

Shokarev

2023

Ufimsk. Mat. Zh.

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In the paper we solve the problem on the representability of a fourth order ordinary differential equation in the form of Hamilton-Ostrogradsky equations. Local bilinear forms play an essential role in the investigation of the potentiality property of the considered equation. It is well known that the problem of representing differential equations in the form of Hamilton-Ostrogradsky equations is closely related to the existence of a solution to the inverse problem of the calculus of variations, that is, for a given equation one needs to construct a functional-variational principle. To solve this problem, we first obtain necessary and sufficient conditions for the given equation to admit an indirect variational formulation relative to a local bilinear form and then construct the corresponding Hamilton-Ostrogradsky action. Note that the found conditions are analogous to the Helmholtz potentiality conditions for the given ordinary differential equation. We also define the structure of the considered equation with the potential operator and use the Ostrogradsky scheme to reduce the given equation to the form of Hamilton-Ostrogradsky equations.It should be noted that applications and extensions of the work are the possibility to establish connections between invariance of the functional and first integrals of the given equation and to extend the proposed scheme to partial differential equations and systems of such equations.

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Section: Introductionmentioning

confidence: 99%

On indirect representability of fourth order ordinary differential equation in form of Hamilton - Ostrogradsky equations

Budochkina,

Luu,

Shokarev

2023

Ufimsk. Mat. Zh.

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“…These problems are also related to the mechanics of finite-and infinite-dimensional systems [7,8,[11][12][13]. There is a large number of works devoted to IPCVs for different types of equations and their systems: in particular, for ordinary differential equations and differential equations with partial derivatives [4,6,13,18,19,21], operator equations [2,3,14,15], differentialdifference equations [5,9,10], and stochastic differential equations [16,17]. In these works, nonlocal bilinear forms were mainly used to solve an IPCV.…”

Section: Introductionmentioning

confidence: 99%

On the Potentiality of a Class of Operators Relative to Local Bilinear Forms

Budochkina

Dekhanova

2021

UMJ

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The inverse problem of the calculus of variations (IPCV) is solved for a second-order ordinary differential equation with the use of a local bilinear form. We apply methods of analytical dynamics, nonlinear functional analysis, and modern methods for solving the IPCV. In the paper, we obtain necessary and sufficient conditions for a given operator to be potential relative to a local bilinear form, construct the corresponding functional, i.e., found a solution to the IPCV, and define the structure of the considered equation with the potential operator. As a consequence, similar results are obtained when using a nonlocal bilinear form. Theoretical results are illustrated with some examples.

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“…To find the first integrals by means of the variational symmetries one has to study the question on existence of the action functional, that is, to solve the inverse problem of the calculus of variations including that for the equations with non-potential operators. The construction of direct and indirect variational formulations for various types of equations and systems were studied, for instance, in works [4], [5], [6], [7], [8], [9], [10], [11], [12]. In works [13], [14], there was established a relation between the symmetries of Euler and non-Euler functionals with the first integrals of the corresponding motion equations.…”

Section: Introductionmentioning

confidence: 99%

On connection between variational symmetries and algebraic structures

Budochkina

2021

Ufimsk. Mat. Zh.

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In the work we present a rather general approach for finding connections between the symmetries of -potentials, variational symmetries, and algebraic structures, Lie-admissible algebras and Lie algebras. In order to do this, in the space of the generators of the symmetries of the functionals we define such bilinear operations as (S, T)-product, G-commutator, commutator. In the first part of the work, to provide a complete description, we recall needed facts on -potential operators, invariant functionals and variational symmetries. In the second part we obtain conditions, under which (S, T)-product, G-commutator, commutator of symmetry generators of -potentials are also their symmetry generators. We prove that under some conditions (S, T)-product turns the linear space of the symmetry generators of -potentials into a Lie-admissible algebra, while Gcommutator and commutator do into a Lie algebra. As a corollary, similar results were obtained for the symmetry generators of potentials, ≡ , where the latter is the identity operator. Apart of this, we find a connection between the symmetries of functionals with Lie algebras, when they have bipotential gradients. Theoretical results are demonstrated by examples.

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On the existence of variational principles for differential-difference evolution equations

Cited by 4 publications

References 3 publications

On indirect representability of fourth order ordinary differential equation in form of Hamilton - Ostrogradsky equations

On indirect representability of fourth order ordinary differential equation in form of Hamilton - Ostrogradsky equations

On the Potentiality of a Class of Operators Relative to Local Bilinear Forms

On connection between variational symmetries and algebraic structures

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