euler-lagrange equations

In this paper, we aim to extend the Darboux frame field into 3-dimensional Anti-de Sitter space and obtain two cases for this extension by considering a parameterized curve on a hypersurface; then we carry out the Euler-Lagrange equations and derive differential equations for non-null elastic curves in AdS$_{3}$ (i.e. 3-dimensional Anti-de Sitter space). In this study, we investigate the elastic curves in AdS$_{3}$ and obtain equations through which elastic curves are found out. Therefore, we solve these equations numerically and finally plot and design some elastic curves.

The focus of this article is on the study of discrete optimal control problems (DOCPs) governed by time-varying systems, including time-varying delays in control and state variables. DOCPs arise naturally in many multi-stage control and inventory problems where time enters discretely in a natural fashion. Here, the Euler--Lagrange formulation (which are two-point boundary values with time-varying multi-delays) is employed as an efficient technique to solve DOCPs with time-varying multi-delays. The main feature of the procedure is converting the complex version of the discrete-time optimal control problem into a simple form of differential equations. Since the main problem is in discrete form, then the Euler--Lagrange equation changes to an algebraic system with initial and final conditions. The graphic representation of numerical simulation results shows that the proposed method can effectively and reliably solve DOCPs with time-varying multi-delays.

Abstract. Let G be a Banach Lie group modeled on the Banach space, possibly infinite dimensional, E. In this paper first we introduce Euler-Lagrange equations on the Lie group G with potential and right invariant metric. Euler-Lagrange equations are natural extensions of the geodesic equations on manifolds and Lie groups. In the second part, we study the geometry of the mechanical system of a rigid body with a fixed point in the gravitational field. This Mechanical systems is usually know as symmetric heavy top. Then we show that the extracted equations by this theory coincide with the known equations of heavy top. Finally, as an infinite dimensional example, we study the Camassa-Holm equations on Bott-Virasoro group at the presence of potential. Bott-Virasoro group is the product of the group of diffeomorphisms of the circle of Sobolev class by the real line and by a potential on a Lie group G we mean a differentiable function from G to the real line R.