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Journal of Physics Conference Series

Lagrangians and Hamiltonians with friction

C E Smith

Published Jun 1, 2010• 10 citations
Publisher

Abstract

Methods are presented for extending a promising formalism that incorporates dissipative forces into Quantum Mechanics. Hamiltonians and Lagrangians are constructed that have additional force and frictional terms while still being consistent with the rules of the formalism. The simple systems of a damped particle in a force field and a quadratically damped oscillator are discussed.

DOI
10.1088/1742-6596/237/1/012021

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Methods are presented for extending a promising formalism that incorporates dissipative forces into Quantum Mechanics. Hamiltonians and Lagrangians are constructed that have additional force and frictional terms while still being consistent with the rules of the formalism. The simple systems of a damped particle in a force field and a quadratically damped oscillator are discussed.
A Lagrangian that yields equations of motion for a damped simple harmonic oscillator is shown not to describe this system, but a completely different physical system. Thus, we have a simple counterexample to the statement: ’’If a Lagrangian gives the correct equations of motion for a given system then the Lagrangian describes the system.’’ We also construct a physical system that the Lagrangian describes and derive some of its properties. The quantum theory based on the above Lagrangian has been used to attempt to discuss the quantum theory of dissipation. This is, however, incorrect since the system the Lagrangian describes has no dissipation.
Phenomenological Equations of Motion for Dissipative Systems Lagrangian Hamiltonian and Hamilton-Jacobi Formulation of the Classical Dissipative Systems Noether's Theorem and Non-Noether Conservation Laws Dissipative Forces Derived from Many-Body Problems A Particle Coupled to a Field and the Damped Motion of a Central Particle Coupled to a Heat Bath Quantization of Dissipative Systems in General and of Explicitly Time-Dependent Hamiltonians in Particular Density Matrix and the Wigner Distribution Function for Damped Systems Path Integral Formulation of a Damped Harmonic Oscillator Quantization of the Motion of an Infinite Chain Heisenberg's Equations of Motion for a Particle Coupled to a Heat Bath Quantum Mechanical Models of Dissipative Systems and the Concept of Optical Potential.
This thesis examines dissipative forces in physical systems in both microscopic and quantum level. The system-environment interaction is modeled by assuming that the environment is composed of simple harmonic oscillators and that there is a linear interaction between these oscillators and the system. The classical case is solved by determining the retarded Green’s function, which yields the equation of motion of the system. For quantum dissipation, the Lindblad master equation is used to describe the time evolution of the reduced density matrix of the system, and the propagator is investigated to identify important parameters, including the term of Newton's friction force.