Research Papers

The Extrema of an Action Principle for Dissipative Mechanical Systems

[+] Author and Article Information
Qiuping A. Wang

e-mail: awang@ismans.fr
Laboratoire de Physique Statistique et
Systemes Complexes,
LUNAM Université,
44, Avenue, F.A. Bartholdi,
Le Mans 72000, France;
UMR CNRS 6283,
Université du Maine,
Le Mans 72085, France

The usual variational calculus with Maupertuis action is as follows:Display Formula


Substitute dx=x·dt for dx, the first term becomes δ(p2/2m)dt and the second term becomes -mx··δxdt through the time integration by part of δx· under the condition δx(a)=δx(b)=0. The total energy conservation at the moment t of the variation means δH=δ(p2/2m)+(V/x)δx+(Ed/x)δx=0 or δ(p2/2m)=-(V/x)δx-(Ed/x)δx. Finally, we haveDisplay Formula


which implies that the Maupertuis principle δAM=0 necessarily leads to the Newtonian equation of damped motionDisplay Formula


where we have usedDisplay Formula


according to the second fundamental theorem of calculus.

Manuscript received September 21, 2012; final manuscript received April 12, 2013; accepted manuscript posted May 29, 2013; published online September 18, 2013. Assoc. Editor: Martin Ostoja-Starzewski.

J. Appl. Mech 81(3), 031002 (Sep 18, 2013) (8 pages) Paper No: JAM-12-1455; doi: 10.1115/1.4024671 History: Received September 21, 2012; Revised April 12, 2013; Accepted May 29, 2013

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 a numerical simulation of the damped motion and to compare the actions of different paths in order to obtain 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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Grahic Jump Location
Fig. 1

Illustration of an exaggerated variation operation over the entire optimal path (thick line) from point a to the end point b

Grahic Jump Location
Fig. 2

The ζ dependence of the actions for the optimal path with T = 1 s (for ns = 1000 steps with 10 − 3 s each step). Here, Aop = A0 − Ad is the optimal action (solid line), A0 is the usual action (dashed line), Ad is the dissipative part of the action (dash-dotted line). The drop point ζc can be estimated by ζcT = 1. The inset is an enlarged view of the zone around ζc in the double logarithm plot.

Grahic Jump Location
Fig. 3

The T dependence of the actions for the optimal path with ζ = 1 s−1, where Aop = A0 − Ad is the optimal action (circles), A0 is the usual action (squares), and Ad is the dissipative part of the action (stars)

Grahic Jump Location
Fig. 4

Samples of the different paths created randomly around the optimal path (thick line) given by the solution of Eq. (5) for a small particle moving between two fixed points in linear potential (constant force) and a viscous medium with Stokes' drag constant ζ = 0.1 s−1. The motion lasts T = 1 s with ns = 1000 steps and δt = 10−3 s each step.

Grahic Jump Location
Fig. 5

Illustration of the transition of extrema by comparison of the action of the optimal path (dots) with the actions of other paths (circles) created by the random perturbation of the optimal one. The number of steps is ns = 1000 with δt = 10−3 s each step (T = 1 s). (a) For ζ = 0.1 s−1, (b) for ζ = 1 s−1, and (c) for ζ = 10 s−1. All calculations were made with an amplitude of variation σ = 0.1 mm for a total displacement of about 5 m during T. (d) The ζ dependence of the quantity ΔA = (A¯ - Aop)/(|A¯|+|Aop|), where A¯ is the average action over all of the paths. Here, ΔA can be used to characterize the evolution of extrema of A in three regimes: the minimum regime (ΔA > 0), the maximum regime (ΔA < 0), and the saddle point regime around ΔA = 0 for ζcT ≈ 1.

Grahic Jump Location
Fig. 6

The T dependence of the characteristic value ζc, which decreases with increasing T. It can be approximated by ζcT = 1.

Grahic Jump Location
Fig. 7

The T dependence of the quantity ΔA for ζ = 1 s−1 and δt = 10−3 s. The characteristic point Tc of the evolution can be approximated by Tc = 1/ζ.

Grahic Jump Location
Fig. 8

Comparison of the action of the optimal path (dot) with the actions of the perturbed paths (circles) for three overdamped motions: (a) with the constant conservative force damped by constant friction fd = , where ζ = 9.99 ms−2 and σ = 0.1 nm, (b) with the constant conservative force damped by the quadratic drag fd=mζx·2, where ζ = 1 m−1 and σ = 0.1 mm, and (c) with the harmonic oscillator damped by Stokes' drag, where ζ = 1.1 s−1 and σ = 0.1 mm. The number of steps is ns = 1000 with δt = 10−3 s each step (T = 1 s). In all three cases, the action of the Newtonian path (dot) is a maximum, while it was a minimum for small ζ (not shown here). The ζ-dependent transition of extrema is similar to Fig. 5.



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