The Extrema of an Action Principle for Dissipative Mechanical Systems

Abstract

A least action principle for damping motion has been previously proposed with a Hamiltonian and a Lagrangian containing the energy dissipated by friction. Due to the space-time nonlocality of the Lagrangian, mathematical uncertainties persist about the appropriate variational calculus and the nature (maxima, minima and inflection) of the stationary action. The aim of this work is to make numerical simulation of damped motion and to compare the actions of different paths in order to get evidence of the existence and the nature of stationary action. The model is a small particle subject to conservative and friction forces. Two conservative forces and three friction forces are considered. The comparison of the actions of the perturbed paths with that of the Newtonian path reveals the existence of extrema of action which are minima for zero or very weak friction and shift to maxima when the motion is overdamped. In the intermediate case, the action of the Newtonian path is neither least nor most, meaning that the extreme feature of the Newtonian path is lost. In this situation, however, no reliable evidence of stationary action can be found from the simulation result.

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“…For the time being, this same approach, based on the least action principle, cannot be applied to dissipative systems for which this principle is no more valid. The extension of the least action principle to dissipative system is still under progress [19].…”

Is it possible to formulate least action principle for dissipative systems?

“…With this Lagrangian, the question of the nonlocality in time (see the remarks below) of the variational calculus, raised in the first formulation of the LAP for dissipative systems and discussed in detail in [22], does not arise. The variation calculus can be made in the usual way as with the action of Eq.(1).…”

“…The reason is that the time integral in Eq. ( 8) may be misleading and give a false impression that the energy H 2 and, consequently, the Lagrangian L are nonlocal in space and time [22], while H 2 , in the expression Eq. ( 4) with its proper variables x i (t) and ẋi (t), is completely local in time.…”

Back to Maupertuis' least action principle for dissipative systems: not all motions in Nature are most energy economical

“…In a recent work [25,26], we have proposed a simple and universal Lagrangian for any dissipative force and formulated the HPLA for dissipative systems. The essential of this work is the idea of an isolated (hence Hamiltonian) total system including the damped moving body and its environment, coupled to each other by dissipative force, with a total Hamiltonian composed of the kinetic energy, the potential energy of the body, and the energy lost by the body into the environment due to dissipation.…”

“…The corresponding action is A = t b ta Ldt which has been used for a general formulation of the HPLA for dissipative systems [25]. The extremum property of A was verified by numerical simulation of damped motion in [26].…”