arXiv:1512.05339v1 [cond-mat.stat-mech] 16 Dec 2015 uae yMueti n1744[ A in Maupertuis by mulated action Least ple, nte point another and isptv ytm:ntalmtosi Nature in motions all not systems: dissipative akt aprus es cinpicpefor principle action least Maupertuis’ to Back M h rnil flatato PA a o h rttm lal for- clearly time first the for was (PLA) action least of principle The ewrs lsia ehnc,dsiaiesses aitoa p Variational systems, dissipative mechanics, Classical Keywords: m = id fmtos:alsainr oin ihcntn velo constant dissipation drag. with Stokes least motions by to stationary damped equivalent all motions r all least is : of motions action path of least the kinds follow the mechanic or that classical dissipation of and energy motions least all of not that rule princi shown action is least of it the connection basis, to a Lagr dissipation an the energy and of of s systems timization derivation a dissipative simple any as a for used mechanics allowing variationa be motions, the of damped can formulation for the principle, for action approach powerful least and Maupertuis the ics, h aso oymvn along moving body a of mass the R a b ti hw hta letfr fvrainlcluu fmec of calculus variational of form oldest an that shown is It pdx b = nveOiirMsia,705L as France Mans, Le 72085 Messiean, Olivier Anevue MM M NS68,Uiested Maine, Universit´e du 6283, CNRS UMR IMMM, r oteeg economical energy most are ttime at R t t a b p xdt ˙ t b where ecle i rnil eahsclprinciple metaphysical a principle his called He . ipn .Wang A. Qiuping 1 p Abstract ihtefloigdfiiino naction an of definition following the with ] = 1 m x ˙ x stemmnu,˙ momentum, the is xsfo point a from axis ls nthis On ples. bythe obey s principle l a esistance, x iyand city ttime at o two for h op- the angian h velocity the imple han- rinci- t a to because he thought that the space integral of the momentum along the tra- jectory of the motion, AM , represents the effort or the cost of the production of the motion1 which the intelligence of the Nature must minimize. There was no explanation why pdx is the effort or the cost of a motion. Thereafter, Euler applied this principle to mechanical motion[17] with explicit use of the condition of energy conservation which Maupertuis has also implicitly used in his 1746’s paper for elastic collision. Nowadays, it is a consensus that Maupertuis principle applies to energy conservative systems, i.e., the motion does not cost anything in term of energy. Clearly, the statement that AM is the cost of a motion does not hold. Today it is well understood that the Maupertuis principle of least action (MPLA) is equivalent to the Hamilton principle of least action (HPLA), each of them having different constraints of variational calculus[3, 4]. The Legendre transformation L = px˙ − H implies A = AM − T H¯ where L is the Lagrangian, H the Hamiltonian, T = ta − tb the duration of the motion, tb ¯ 1 tb A = ta Ldt the action, and H = T ta Hdt the time average of H. Then it ¯ ¯ is straightforwardR to write the variationR relation δA + HδT = δAM − T δH, and to derive Newtonian equation either by the MPLA δAM = 0 under the condition of constant energy δH = 0, or by the HPLA δA = 0 under the condition of constant duration of motion δT = 0[5, 6, 7]. The subsequent development of the PLA has largely favored HPLA which has been successfully applied not only to the entire classical physics but also to the quantum physics within the path integral formalism[9]. This success has given to HPLA a conceptual priority to all other principles, empirical laws and differential equations in different branches of physics, and inspired three major projects of its extension. The first one is to deepen the understanding of nature through this principle and to search for the fundamental meaning of its exceptional universality in physics[6, 8, 7]. The second one is to extend it to more domains such as thermodynamics, statistical mechanics (with the pioneer effort, though unfruitful, of Boltzmann, Helmholtz and Hertz[10]), large deviation theory[11] and stochastic mechanics[12]. The third one is to formulate it, within classical mechanics, for damped motion of dissipative systems[3, 13, 14, 15, 16]. Dissipative motion has been treated for the first time by Euler [17, 18] in the formulation of the calculus of variation for Brachistochrone problem

1C’est cela, c’est cette quantit´ed’action qui est ici la vraie d´epense de la Nature, et ce qu’elle m´enage le plus qu’il est possible dans le mouvement de la lumi`ere [1].

2 with friction, the latter being written as a nonlinear function of the square of the velocity whose physical sense is not explicit. This effort was followed by Rayleigh[19], with more physical consideration, in the proposition of a 1 2 ‘dissipative function’ D = 2 ζx˙ , for the special case of the Stokes’ law with d ∂L ∂D ∂L the drag force fd = −mζx˙, to write dt ∂x˙ − ∂x˙ − ∂x = 0, where ζ is the drag constant and m the mass of the damped body. Nevertheless, this ‘dissipative function’ does not have any relationship with the energy dissipated by the drag force. The subsequent efforts during a long period [14, 19] have led to many ‘dissipative Lagrangian function’ [16, 20, 21, 22, 23, 24] all suffering from the shortcomings such as the non uniqueness, non universality, absence of clear physical meaning and of close energy connection like L = K − V [3, 13, 14]. In a recent work [25, 26], we have proposed a simple and universal La- grangian for any dissipative force and formulated the HPLA for dissipative systems. The essential of this work is the idea of an isolated (hence Hamilto- nian) total system including the damped moving body and its environment, coupled to each other by dissipative force, with a total Hamiltonian com- posed of the kinetic energy, the potential energy of the body, and the energy lost by the body into the environment due to dissipation. For simplicity, we suppose that the 1D body (system 1) is large with respect to the particles of the environment, and that it moves along the axis x with velocityx ˙. Its environment (system 2 composed of N particles with positions xi and veloc- itiesx ˙ i and i = 1, 2, ..., N ) includes all the parts coupled to system 1 by friction and receiving the dissipated mechanical energy. The energy transfer from system 1 to system 2 occurs only through a friction force. The total 1 2 Hamiltonian reads H = K1 + V1 + K2 + V2 + Hint where K1 = 2 mx˙ is the 1 N 2 kinetic energy and V1 the potential energy of the system 1, K2 = 2 i mix˙ i the kinetic energy and V2(x1, x2, ...xN ) the potential energy of systemP 2, and Hint the interaction energy between the system 1 and 2. Hint is responsi- ble for the friction law and determined by the coupling mechanism on the interface between the moving body and the environment. We can suppose that the coupling mechanism, the interface (body’s shape and size, body- environment distance, nature of the closest parts of the environment to the interface, etc.) and the friction laws do not change with the virtual variation of path of the damped body. In this case, Hint is a constant of variation and can be neglected. The Hamiltonian is then H = K1 + K2 + V1 + V2 or H = H1 + H2 where H1 = K1 + V1 is the total energy of the system 1 and

3 H2 = K2 + V2 the total energy of the system 2 :

N 1 2 H2 = mx˙ i (t)+ V2[x1(t), x2(t)...xN (t)] (1) i 2 X a On the other hand, the energy of system 2 can be written as H2 = H2 + Ed a where H2 is its energy (a constant) at ta and Ed the energy dissipated from system 1 to 2 up to a time moment t (ta ≤ t ≤ tb). Eb is given by the work ~ of the friction force fd in the following way:

x(t) Ed = − fd(τ)dx(τ) (2) Zxa where τ is any time moment between ta = 0 and t. According to the second ∂ x(t) fundamental theorem of calculus[28] F (x(t)) = ∂x(t) xa(0) F (τ)dx(τ), we get ∂Ed ∂H2 fd(t) = − ∂x(t) = − ∂x(t) . Although this relation looksR like the relation fc = ∂V1 − ∂x(t) for the conservative force fc acting on the system 1, Ed or H2 cannot be considered as a potential energy of the system 1 for the following reasons: on the one hand, it depends on the history of the motion, on the other, it cannot be directly and completely converted back into kinetic energy of the system 1. The Lagrangian L of the whole system can be defined by using the Leg- endre transformation [25, 26]:

N L = px˙ + pix˙ i − H = K1 − V1 +2K2 − H2 (3) i X where p is the momentum of system 1 and pi the momentum of the particle tb i of the system 2. The corresponding action is A = ta Ldt which has been used for a general formulation of the HPLA for dissipativeR systems [25]. The extremum property of A was verified by numerical simulation of damped motion in [26]. One of the aims of the formulation of PLA for dissipative systems is to study the connection between PLA and the extremum property of energy dissipation such as the maximum or minimum dissipation, least distance or least resistance[13, 14, 27]. However, with the formulation of HPLA men- tioned above, we do not see clear hint to this connection. This is why we have thought of another form of PLA : the Maupertuis principle of least action and its formulation for dissipative motion. In what follows, we will

4 show first of all how to make the calculus of variation with the AM defined for the whole conservative system :

tb N AM = [px˙ + pix˙ i]dt. (4) ta i Z X

The second term with the summation in the integral of AM being independent of x, the variation of AM due to a tiny change δx(t) of the path of the damped body is given by

tb tb δAM = δ pxdt˙ = (xδp ˙ + pδx˙)dt. (5) Zta Zta 2 tb p tb where the first term is just ta δ( 2m )dt = ta δK1dt and the second term tb tb tb dp tb becomes ta pδxdt˙ = pδx|ta −R ta dt δxdt = −R ta mxδxdt¨ with the condition δx(a)= δxR (b) = 0. Eq.(5) nowR reads R

tb δAM = [δK1 − mxδx¨ ]dt. (6) Zta

Now we introduce the constraint of conservation of total energy δH = δH1 + δH2 = δK1 + δV1 + δH2 = 0 or δK1 = −δV1 − δH2 = 0. Eq.(5) becomes tb δAM = ta [−δV1 − δH2 − mxδx¨ ]dt, or R tb ∂V1 ∂H2 δAM = − − − mx¨ δxdt (7) Zta " ∂x ∂x # which implies that the Maupertuis principle δAM = 0 necessarily leads to the Newtonian equation of damped motion: ∂(V + H ) mx¨ = − 1 2 = f + f . (8) ∂x c d This is a general formulation of MPLA for damped motion subject to any friction force. The stationarity δAM = 0 must be a minimum since the integral of AM in Eq.(4), by definition, does not have upper limit. We can also calculate the 2 tb 2 tb 2 second variation δ AM = ta δ pxdt˙ = 2m ta δx˙ dt ≥ 0, which proves the minimum of the vanishingR first variation δARM = 0. As expected, this dissipative MPLA makes it possible to study easily the connection between PLA and the optimization of energy dissipation. Let us

5 first show an interesting case where the motion is damped by Stokes drag fd = −mζx˙. The variation of the Maupertuis action can be written as

xb b 1 δEd δAM = δ mζxdx˙ = . (9) ζ Zxa ζ b xb where Ed = xa ζxdx˙ is the quantity of the work of the friction force over the b entire trajectoryR from a to b. Hence the MPLA δAM = 0 entails δEd = 0, i.e., least dissipation. In other words, the path of least action is just the path of least resistance. The connection between MPLA and optimization of dissipation can be investigated in a more general way. Let fd = −f(x ˙) be a certain friction force where f(x ˙) is a positive increasing function of the magnitude of velocity. b xb xb Let Ed = − xa fddx = xa f(x ˙)dx be the dissipated energy over the entire b trajectory fromR a to b. ChangingR the integral variable into time gives Ed = T T 0 f(x ˙) |x˙| dt = 0 P (x ˙)dt where P (x ˙)= f(x ˙) |x˙| is the power of the friction forceR to do workR and should be always positive and increasing function of |x˙|. We can write P (x ˙ 2) = P (y) with y =x ˙ 2. The variations δx and δx˙ yield a a variation of Eb as follows

xb xb b dP (y) dP (y) δEd = 2xδ ˙ xdt˙ = (xδ ˙ x˙ +xδ ˙ x˙)dt (10) Zxa dy Zxa dy We write the first term in the parentheses as fllows P ′(y)xδ ˙ x˙ = P ′(y)δ( 1 x˙ 2)= ′ 2 P (y) ∂V1 ∂Ed ′ dP (y) − m ( ∂x + ∂x )δx where P (y)= dy . We have used the conservation of total energy δH = δ(K1 +V1 +Ed) = 0. For the second term, we make a time integral by parts and use the same conditions as in the passage from Eq.(5) ′ T ′ T d ′ T dP (y) to Eq.(7) to write 0 P (y)xδ ˙ xdt˙ = − 0 dt [P (y)x ˙]δxdt = − 0 [ dt x˙ + ′ P (y)¨x]δxdt. Finally,R we have R R T ′ b dP (y) ′ ∂V1 ∂Ed δEd = − mx˙ + P (y)( + + mx¨) δxdt. (11) Z0 " dt ∂x ∂x #

∂V1 ∂Ed Since mx¨ = − − as a consequence of the MPLA δAM = 0, we see that ′∂x ∂x b T dP (y) b δEd = − 0 dt mxδxdt˙ which is not vanishing, in general. So Ed does not have stationarityR on the least action path or, in other words, this latter is not necessarily the path of least dissipation or resistance. This is our general result. ′ dP (y) b However, it is clear from Eq.(11) that if dt = 0, δEd vanishes whenever ∂V1 ∂Ed δAM = 0 with mx¨ = − ∂x − ∂x . There are many motions satisfying this

6 condition. The first one is the Stokes’ drag giving P = mζx˙ 2 = mζy and dP ′(y) d(mζ) dt = dt = 0. Since P (y) is an increasing function of the magnitude of ′ dP (y) b velocity or of y, P (y) = dy is always positive, so δEd = 0 is a minimum just as δAM = 0. This is in accordance with the conclusion obtained above with Eq.(9). dP ′(y) The second motion leading to dt = 0 is the motion with constant velocity, be it along a straight or curved path and whatever is the velocity dependence of the power P . From Eq.(11), we still have the equivalence b between MPLA and least dissipation, i.e., δEd = 0 following δAM = 0. This conclusion can also be reached from the following analysis. By definition, the b action AM = a pds, (dx is replaced by ds for curved path where s = f(t) b is the movingR equation). The result is AM = pLab where Lab = a ds is the length of the path. Hence MPLA δAM = 0 entails δLab = 0, i.e.,R the path of least distance. Since the velocity is constant, least distance implies least b time δT = 0 where T = a dt is the duration of the motion. On the other hand, as the friction forceR is constant, the energy dissipated between a and b b b is Ed = fd a ds = fdLab. The minimum distance δLab = 0 then yields a b minimum dissipationR δEd = 0. In summary, by considering a conservative system composed of the mov- ing body and its environment coupled by friction, we have formulated a Maupertuis principle of least action for damped motion. This formalism al- lows to connect the optimization of energy dissipation to PLA. According to this formulation, the Maupertuis principle of least action is equivalent to the least dissipation or path of least resistance in the following two cases: 1) all motions damped by Stokes drag, and 2) all stationary motions with constant velocity. Otherwise, the paths of least action (solutions of the Newtonian equation) are not necessarily the least dissipative in energy. Since the New- tonian laws are universal for classical mechanical systems, this work means that the extremum rules such as least dissipation, least resistance, least ef- fort, or maximum dissipation are not general laws for mechanical motions. However, the above two cases include a very large number of mechanical motions in Nature for which the Maupertuis action AM really represents the energy cost of motion. Mr. Maupertuis was partially right.

7 References

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