A geometric description for simple and damped harmonic oscillators

Zahide Ok Bayrakdar,Tuna Bayrakdar

doi:10.3906/mat-1902-73

Abstract

In this work we consider the Riemannian geometry associated with the differential equations of one dimensional simple and damped linear harmonic oscillators. We show that the sectional curvatures are completely determined by the oscillation frequency and the friction coefficient and these physical constants can be thought as obstructions for the manifold to be flat. Moreover, equations of simple and damped harmonic oscillators describe nonisomorphic solvable Lie groups with nonpositive scalar curvature.

Highlights

An oscillatory motion of a massive particle of one degree of freedom is governed by second-order linear ordinary differential equation (ODE) d2x dt2 + γ dx dt ω02x = F (t), (1.1)here ω0 is angular frequency, γ is damping coefficient and F (t) is a time-dependent external force exerted on the system
It is possible to distinguish them by considering such an equation as a free particle equation due to the curvature of Riemannian structure constructed by the coframe associated with given ODE and scalar curvature is completely interpreted in terms of angular frequency and damping coefficient
In this paper we deal with the Riemannian geometry associated with the equations for one dimensional linear harmonic motions

Summary

Introduction

An oscillatory motion of a massive particle of one degree of freedom is governed by second-order linear ordinary differential equation (ODE). Our aim is to uncover the geometric meaning of physical constants angular frequency and damping coefficient By this means, we consider the equations of harmonic motions in the framework of Riemannian geometry associated with second-order ODEs. 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. From ∇e1 e1 = 0 we see that once a second-order ODE is given one can constructed unique torsion free metric compatible connection on the tangent bundle of a submanifold by means of the exterior differential system (2.3) such that 2-jet of a solution curve defines a geodesic curve on the manifold corresponding to a second-order ODE. Corollary 2.2 The 2-jet of a solution curve of equation (2.1) is a geodesic curve on the Riemannian manifold S

Lie group structure of harmonic motions

Damped harmonic oscillator

Concluding remarks