Minimal Surfaces in Three-Dimensional Riemannian Manifold Associated with a Second-Order ODE

A geometric description for simple and damped harmonic oscillators

“…This work can be seen as the continuation of the recent work of the authors dealing with a first order differential equation as a curved space in jet manifold in the context of Riemannian geometry [26]. Our consideration in this paper is based on constructing Riemannian metric and connection 1-form from the exterior differential system encoding a second-order ODE in the secondorder jet space [2]. By this approach it is seen that 2-jet of a solution curve of a given equation defines a geodesic curve of the metric compatible connection.…”

“…As it is considered in [2,26], an ODE can be treated as a submanifold of an appropriate jet space and Riemannian structure may be given by the sum of squares of the elements of the exterior differential system encoding given differential equation. Our aim in this paper is to uncover the geometric role of the physical quantities oscillation frequency and damping coefficient in equations of harmonic motions.…”

“…Our aim in this paper is to uncover the geometric role of the physical quantities oscillation frequency and damping coefficient in equations of harmonic motions. In order to deal with different types of oscillatory motions, we mainly follow [2] and consider the general second-order ordinary differential equation of the form…”

“…To see this, we write these equations in the vector form as [e i , e j ] = −c i jk e k to obtain respectively. A Lie algebra g is said to be solvable if the derived series g (0) = g, g (1) = [g, g], g (2) = [g (1) , g (1) ], g (3) = [g (2) , g (2) ], • • • vanishes for some k > 0 , i.e. g (k) = 0.…”

Geometry of submanifolds of all classes of third-order ODEs as a Riemannian manifold

“…The main idea for writing present paper is raised from the papers [5], written by T. Bayrakdar and A. A. Ergin.…”