Invariance of functionals and related Euler–Lagrange equations

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

doi:10.3103/s1066369x17020062

Cited by 11 publications

(4 citation statements)

References 3 publications

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“…It was shown in monograph [17] that the symmetries of Euler functionals are also symmetries of the corresponding Euler-Lagrange equations. In works [18], [19], similar results were obtained in the general case for non-Euler functionals, to which equations with quasi-potential operators correspond. A role of algebraic structures associated with motion equations is well-known in the mechanics of finite-dimensional and infinite-dimensional systems [6], [7], [17], [20], [21], [22], [23].…”

Section: Introductionsupporting

confidence: 61%

“…In work [7], there was studied the invariance with respect to the divergence of the generalized in the Pfaff sense action, a formula for finding the first integrals of the operator Birkhoff equation was obtained and it was proved that the generators of the divergent symmetries of the functional form a Lie algebra with respect to the commutator. These studies were continued in works [13], [14], [18], [19]. Moreover, in work [24], there were obtained the conditions under which the (S, T)-product, the G-commutator, the commutator of the symmetry generators for the operator equations are also the symmetry generators and a relation between the symmetries of operator equations and Lie-admissible algebras and Lie algebras was found.…”

Section: Introductionmentioning

confidence: 97%

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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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Section: Introductionsupporting

confidence: 61%

Section: Introductionmentioning

confidence: 97%

On connection between variational symmetries and algebraic structures

Budochkina

2021

Ufimsk. Mat. Zh.

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“…The monograph is devoted to the problem of representing a second-order ODE in the form of the Lagrange, Hamilton and Birkhoff equations. In [22][23][24], methods for solving the Helmholtz problem are developed for partial differential equations (PDEs). In [20,21,25], the authors present their studies on the Helmholtz problem, mainly for ODEs and PDEs, as well as a historical overview of the development and generalization of the problem.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Helmholtz problem and convergence in distribution

Tleubergenov

Vassilina

Azhymbaev

2022

Filomat

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In the present paper, the solvability of the stochastic Helmholtz problem is investigated in the class of stochastic differential equations equivalent in distribution. Earlier, by additional variables method the Helmholtz problem was investigated in the class of stochastic differential equations equivalent almost surely (a.s.). The study of the stochastic Helmholtz problem in the class of equations equivalent in distribution allows us to significantly expand the region of its solvability. This is due to the possibility of using well-known methods of the theory of stochastic processes, such as the method of the phase space transformation, the method of absolutely continuous change of measure, and the method of random change of time. In that paper stochastic equations of the Lagrangian structure equivalent in distribution are constructed by the given second order Ito stochastic equations using the methods of phase space transformation, absolutely continuous measure transformation and randomtime substitution. The obtained results are illustrated by specific examples.

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“…Inverse problems of dynamics in the class of partial differential equations are studied in [12][13][14], and in the class of stochastic differential equations in [15][16][17][18][19].…”

mentioning

confidence: 99%

On Inverse Stochastic Reconstruction Problem

Tleubergenov¹,

Vassilina²,

Sarypbek³

2021

News of NAS RK. SPh-M

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In this paper, general reconstruction problem in the class of second-order stochastic differential equations of the Ito type is considered for given properties of motion, when the control is included in the drift coefficient. And the form of control parameters is determined by the quasi-inversion method, which provides necessary and sufficient conditions for existence of a given integral manifold. Further, the solution of the Meshchersky’s stochastic problem is given, which is one of the inverse problems of dynamics and, according to the well-known Galiullin’s classification, refers to the restoration problem. It is assumed that random perturbations belong to the class of processes with independent increments. To solve the posed problem an equation of perturbed motion is drawn up by the Ito rule of stochastic differentiation. And, further, the Erugin method in combination with the quasi-inversion method is used to construct: 1) a set of control vector functions and 2) a set of diffusion matrices that provide necessary and sufficient conditions for a given second-order differential equation of Ito type to have a given integral manifold. The linear case of a stochastic problem with drift control is considered separately. In the linear setting, in contrast to the nonlinear formulation, the conditions of solvability in the presence of random perturbations from the class of processes with independent increments coincide with the conditions of solvability in a similar linear case in the presence of random perturbations from the class of independent Wiener processes. Also considered is the scalar case of the recovery problem with drift controls.

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Invariance of functionals and related Euler–Lagrange equations

Cited by 11 publications

References 3 publications

On connection between variational symmetries and algebraic structures

On connection between variational symmetries and algebraic structures

Stochastic Helmholtz problem and convergence in distribution

On Inverse Stochastic Reconstruction Problem

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