An operator equation with the second time derivative and Hamiltonian-admissible equations

An operator equation with the second time derivative is represented in the form of a Hamiltonianadmissible equation. A relationship between solutions of Hamiltonian-admissible equations and the corresponding Hamiltonian equations is established.     

Variational symmetries of Euler and non-Euler functionals

We develop a unified approach to the investigation of invariant properties of Euler and non-Euler functionals and establish a relationship of variational symmetries with first integrals of a given evolution operator equation of second order with respect to t. In addition, we investigate the properties of the generators of divergence symmetries.     

Invariance of functionals and related Euler–Lagrange equations

We establish a connection between symmetries of functionals and symmetries of the corresponding Euler–Lagrange equations. A similar problem is investigated for equations with quasi-B _u -potential operators.     

On the existence of a variational principle for an operator equation with second derivative with respect to “ time”

Using methods of nonlinear functional analysis, we define the structure of an evolution operator equation of second order that can be formulated in direct variational terms.     

Lie-Admissible Algebras Associated with Dynamical Systems

We introduce the general structures of Lie-admissible algebras in the spaces of Gâteaux differentiable operators and establish their connection with the symmetries of operator equations and the mechanics of infinite-dimensional systems.

     

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

The problem of existence of variational principles for wide classes of generally nonlinear differential-difference equations with nonpotential operators is investigated.