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arXiv:physics/0601133

Expressions for frictional and conservative force combinations within the dissipative Lagrange-Hamilton formalism

Charles Edward Smith

Published Jan 18, 2006• 1 citation
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Abstract

Dissipative Lagrangians and Hamiltonians having Coulomb, viscous and quadratic damping,together with gravitational and elastic terms are presented for a formalism that preserves the Hamiltonian as a constant of the motion. Their derivations are also shown. The resulting L's and H's may prove useful in exploring new types of damped quantum systems.

DOI
10.48550/arxiv.physics/0601133

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Dissipative Lagrangians and Hamiltonians having Coulomb, viscous and quadratic damping,together with gravitational and elastic terms are presented for a formalism that preserves the Hamiltonian as a constant of the motion. Their derivations are also shown. The resulting L's and H's may prove useful in exploring new types of damped quantum systems.
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Some recent works on the path-integral formulation of nonconservative forces quadratic in velocity are examined critically. It is argued that the ambiguity resulting from ordering the operators in the Lagrangian and the Hamiltonian for this force is more serious than it is usually believed, and that one can construct a number of Hermitian Hamiltonians all satisfying Ehrenfest's theorem, but forming a set of noncommuting operators. A similar problem is encountered in the ordering of the Lagrangian for the path-integral approach. This is in addition to a number of well-known difficulties such as violation of the position-velocity uncertainty, and the differences between the invariances of the equation of motion, and the Lagrangian. An alternative way is suggested in which the damping force is replaced by a purely time-dependent term which produces the same classical motion, thus avoiding the above-mentioned difficulties. Using this formulation, wave functions and time-dependent energies are found for a quadratically damped harmonic oscillator, and for a particle moving in a dissipative medium when the friction is proportional to an arbitrary power of velocity.
Two types of Hamiltonians are investigated which describe quantum mechanically a particle moving subject to a linear viscous force under the influence of a conservative force: the conventional explicitly time-dependent one and an alternative class of nonlinear Hamiltonians. In the latter group we propose a new form. By Ehrenfest’s theorem the expectation values of the operators of physical observables correspond to the classical quantities. For all Schrödinger equations we derive and discuss wavepacket, wave, stationary, and pseudostationary solutions of force free motion, free fall, and harmonic oscillator.
For a free falling particle moving in a media which has quadratic velocity force effect on the particle, two equivalent constants of motion, with units of energy, two Lagrangians, and two Hamiltonians are deduced. These quantities describe the dynamics of the same classical system. However, their quantization and the associated statistical mechanics (for an ensemble of particles) describe two completely different quantum and statistical systems. This is shown at first order in the dissipative parameter.