The Derivation of the Equation of the Damped Harmonic Oscillator from Lagrangian Methods with a Kaluza-Klein-like Hidden Dimension

Lagrangian methods cannot include non-conservative forces and therefore cannot produce a simple derivation of the ubiquitous and very useful damped harmonic oscillator equation. Recently, it has been found that the damping term can be included in a Lagrangian approach by employing a Kaluza-Klein hidden dimension (Klein 1926). In this application of the method, the hidden variable occurs as a ``subscale model'' where the normal Lagrangian for a harmonic oscillator is written .5(mq'$^{2}$ -kq$^{2}$ + 2sq'q* +m*q$_{o}$'$^{2}$ ) where q* is an integral over a short cyclic collision time of subscale molecular model coordinate derivative q$_{o}$' where a molecular coordinate shares velocity with a mass m as it collides and ``sticks'' to it, so that q$_{o}$'=q' during the interval of integration, and finally m* is a very small ``molecular'' mass. This results in the total derivative acquiring the term sq' and the resulting equations mq'' +sq' +kq =0 and m*q$_{o}$'=Constant being recovered. This exercise gives insight to the use of Kaluza-Klein hidden dimension in Gravitational physics and also suggests such hidden dimensions can add a ``thermal bath'' to otherwise conservative problems. Klein O. (1926) ``Quantum Theory and Five-Dimensional Relativity Zeitschrift fur Physik'' \textbf{37}, 895.