arXiv subjects

Iulian Duca

Publications and source records attributed to Iulian Duca.

5 recordsLinked to original sources

arXiv2005

Periodical Solutions of Multi-Time Hamilton Equations

To our knowledge, there are two main references [9], [12] regarding the periodical solutions of multi-time Euler-Lagrange systems, even if the multi-time equations appeared in 1935, being introduced by de Donder. That is why, the central objective of this paper is to solve an open problem raised in [12]: what we can say about periodical solutions of multi-time Hamilton systems when the Hamiltonian is convex? Section 1 recall well-known facts regarding the equivalence between Euler-Lagrange equations and Hamilton equations. Section 2 analyzes the action that produces multi-time Hamilton equations, and introduces the Legendre transform of a Hamiltonian together a new dual action. Section 3 proves the existence of periodical solutions of multi-time Hamilton equations via periodical extremals of the dual action, when the Hamiltonian is convex.

math.DS↗

arXiv2005

Some Inequalities Satisfied by Periodical Solutions of Multi-Time Hamilton Equations

The objective of this paper is to find some inequalities satisfied by periodical solutions of multi-time Hamilton systems, when the Hamiltonian is convex. To our knowledge, this subject of first-order field theory is still open. Section 1 recall well-known facts regarding the equivalence between Euler-Lagrange equations and Hamilton equations and analyses the action that produces multi-time Hamilton equations, emphasizing the role of the polysymplectic structure. Section 2 extends two inequalities of [21] from a cube to parallelipiped and proves two inequalityes concerning multiple periodical solutions of multi-time Hamilton equations.

math.DS↗

arXiv2005

Periodical Solutions of Poisson-Gradient Dynamical Systems with Periodical Potential

The main purpose of this paper is the study of the action that produces Poisson-gradient systems and their multiple periodical solutions. The Section 1 establishes the basic tools. The section 2 underlines conditions in which the action $ϕ(u) = \displaystyle\displaystyle\int_{T_{0}}[ \displaystyle% \displaystyle{1/2}| \displaystyle\displaystyle\frac{\partial u}{% \partial t}| ^{2}+F(t,u(t)) ] dt^{1}\wedge >...\wedge dt^{p}$, that produces the Poisson-gradient systems, is continuous, and some conditions in which the general action $ϕ(u) = \displaystyle\displaystyle\int_{T_{0}}L(t,u(t), \displaystyle\displaystyle\frac{\partial u}{\partial t}(t)) dt^{1}\wedge >...\wedge dt^{p}$ is continuously differentiable. The Section 3 studies the multiple periodical solutions of a Poisson-gradient system in the case when the potential function $F$ has a spatial periodicity.

math.DS↗

arXiv2005

Poisson-Gradient Dynamical Systems with Convex Potential

The basic aim is to extend some results and concepts of non-autonomous second order differential systems with convex potentials to the new context of multi-time Poisson-gradient PDE systems with convex potential. In this sense, we prove that minimizers of a suitable action functional are multiple periodical solutions of a Dirichlet problem associated to the Euler-Lagrange equations. Automatically, these are solutions of the associated multi-time Hamiltonian equations.

math.DS↗

arXiv2005

Poisson-Gradient Dynamical Systems with Bounded Non-Linearity

We study the periodical solutions of a Poisson-gradient PDEs system with bounded nonlinearity. Section 1 introduces the basic spaces and functionals. Section 2 studies the weak differential of a function and establishes an inequality. Section 3 formulates some conditions under which the action functional is continuously differentiable. Section 4 analyzes the Poisson-gradient systems and some conditions that ensure periodical solutions.

math.DS↗