About One-Dimensional Conservative Systems with Position Depending Mass

Gustavo López Velázquez,Carlos Rodrigo Martínez Prieto

doi:10.4236/jmp.2014.59093

Abstract

For a one-dimensional conservative system with position depending mass, one deduces consistently a constant of motion, a Lagrangian, and a Hamiltonian for the nonrelativistic case. With these functions, one shows the trajectories on the spaces (x,v) and (x,p) for a linear position depending mass. For the relativistic case, the Lagrangian and Hamiltonian cannot be given explicitly in general. However, we study the particular system with constant force and mass linear dependence on the position where the Lagrangian can be found explicitly, but the Hamiltonian remains implicit in the constant of motion.

Highlights

Position depending mass systems have been relevant since the foundation of the classical mechanics and modern physics [1]-[5]
The interest for these type of problems has grown in modern physics due to fabrication of ultra thin semiconductors [6] [7], inhomogeneous crystals [8], quantum dots [9], quantum liquids [10], and neutrino mass oscillations [11] [12]
We need to mention that this topic is important due to its relation with the foundation of the classical mechanics [13], and its not invariance under Galileo or Poincar-Lorentz transformations [13] [14]

Summary

Introduction

Position depending mass systems have been relevant since the foundation of the classical mechanics and modern physics [1]-[5] (see reference there in). Most of the approaches dealing with position depending mass problems use an intuitive way to write down a Lagrangian or Hamiltonian for the system, and solve the corresponding equations [15] [16]. One obtains a constant of motion, a Lagrangian, and a Hamiltonian in a consistent way for conservative nonrelativistic systems and study the harmonic oscillator with position depending mass as an example. How to cite this paper: Velázquez, G.L. and Martínez Prieto, C.R. (2014) About One-Dimensional Conservative Systems with Posi- tion Depending Mass.

Dynamical Functions

Relativistic Case

Total Force Equal to Zero

Conclusion