Relativistic and . II. A linear translation Hamiltonian-Lagrangian formalism

J G¨u´emez Applied Department University of Cantabria (Spain) December 8, 2020

Abstract A relativistic Hamiltonian-Lagrangian formalism for a composite sys- tem submitted to conservative and non-conservative is developed. A block descending an incline with a frictional , mechanical dissipation process, is described, obtaining an Euler-Lagrange equation including a Rayleigh’s dissipation function. A cannonball rising on an incline, process evolving with mechanical energy production, is described by an Euler-Lagrange equation including a Gibbs’ production function, with a chemical origin force. A matrix four-vector mechanical equation, considering processes’ mechanical and phenomenological aspects, is pos- tulated. This relativistic Hamiltonian-Lagrangian four-vector formalism com- plements the Einstein-Minkowski-Lorentz four-vector fundamental equa- tion formalism. By considering a process’ mechanical and thermody- namic description, temporal evolution equations, relating process’ Hamil- tonian (mechanical energy) evolution and the involved thermodynamic potentials (entropy of the universe, Helmholtz free energy, Gibbs free enthalpy) variations, are obtained.

1 Introduction

For a point particle, m, subjected to a conservative force F , Newton’s second law (NSL in the following) writes F = ma, or F dt = mdv. For a com- posite system, mass M = Σama, constituted by a elements, the set of NSL equations applied to its elements, in general (a > 2), cannot be solved. By applying Newton’s third law, which guarantees that internal forces sum is zero, int ΣaFa = 0 [1, Ch. 1], what is described by Newton’s second law, is the system’s ext centre-of-mass , F = Macm (Newton’s Second-an-a-Half Law [1, ext ext pp. 141-143]), as a function of resultant external force F = ΣaFa , sum of

1 external – conservative and non-conservative – forces applied, with the centre- ext of-mass impulse-linear variation equation F dt = Mdvcm. From NSL applied to the process, for a constant mass process, the so-called comple- ext mentary dynamical relationship [2], dKcm = F dxcm is obtained, relating the 1 2 system’s centre-of-mass variation dKcm = 2 Md(vcm) , to resul- tant external force times centre-of-mass displacement product. For a composite system, the complementary dynamical relationship (a periphrasis to avoid re- ferring to ) equation is essentially a linear momentum equation [3], including, in general, (conservative ) and pseudo-work (non-conservative forces) quantities [4]. The centre-of-mass complementary dynamical relation- ship – also knows as pseudo-work equation [5]– is sometimes mistaken for a ext work-kinetic energy equation, dK = δW [6], (dKcm =6 dK, F dxcm =6 δW ). For a process involving dissipative forces [7], or for a mechanical energy production process [8], the process’ mechanical, and phenomenological, de- scription must be complemented with the laws of thermodynamics to ob- tain a complementary description; in particular, with the first law of ther- modynamics (FLT in the following) relating dKcm, internal energy variations dU, external forces work performed on the system δW ext, and heat δQ, with ext dKcm + dU = δW + δQ [9]. Thermodynamics characterizes each of these, mechanical energy dissipa- tion or mechanical energy production, processes: the entropy of the universe increases when dissipative forces are involved, and mechanical energy produced ultimately comes from some thermodynamic potential decreasing, Helmholtz free energy function or Gibbs free enthalpy function [10]. Process’ thermody- namic potentials (entropy of the universe, Gibbs function) variations calcula- tion allows to characterize its irreversibility, or reversibility [11]. The special , including the of energy principle (relativity is essentially an energy theory, as it is thermodynamics), in its Einstein-Minkowski-Lorentz formalism, allows obtaining the equations, me- chanic and thermodynamic, for a process involving a composite system, with inertia M(ξ, V, T )–ξ chemical composition, V volume, T temperature–, sub- mitted to conservative and non-conservative forces, in a direct and complete way [12]. The four-vector fundamental equation merges two laws, NSL and the FLT, into one relationship. Other relationships, the complementary dynamical relationship or the thermal effects equation, are derived from the four-vector fundamental equation. The Hamiltonian-Lagrangian formalism allows a process’ mechanical de- scription based on Hamilton equations and Euler-Lagrange equation [13, Ch. 6]. An interesting aspect of Hamiltonian-Lagrangian formalism is that its applica- tion to a process requires distinction between forces. A conservative force F is related to a mechanical potential V (x) gradient, with F = −dV (x)/dx. For this formalism to be operational when applied to a composite system, the coordi- nate must be the system’s centre-of-mass , with F = −dV (xcm)/dxcm. For example, for a block descending an incline, one has

p2 H(x , p ) = cm − Mg sin α x , (1) cm cm 2M cm

2 P where centre-of-mass linear momentum is pcm = Mvcm = a mava = p. A dis- sipative force, fR, a friction force, for example, has not a mechanical potential associated with it; it does not enter into the Hamiltonian and has to be consid- ered in a phenomenological way (Amontons-Coulomb kind [14], for example). Conservative and non-conservative forces are incorporated in a similar way into NSL, exercising impulse (time runs) on the system. But a conservative force performs work and derives from a potential energy gradient or work reservoir [15], and a friction force does not perform work on the system and does not come from a mechanical potential (not work reservoir related to it). Force Fξ, coming from a chemical reaction kinetic, has not associated a mechanical po- tential variation but a thermodynamic potential (Gibbs’ free enthalpy function, for example) variation, and does not enter directly into the Hamiltonian [16]. Being always interesting to explore different formalisms to analyze a pro- cess, since relating providing information one can obtain physics insights, in this paper a relativistic Hamiltonian-Lagrangian formalism for processes involv- ing a composite body in linear translation, with mechanical energy dissipation or production, is developed. Processes considered has been previously ana- lyzed in [12] by using a four-vector fundamental equation into an Einstein- Minkowski-Lorentz relativistic mechanics and thermodynamics formulation. Comparing equations obtained by a four-vector fundamental equation (includ- ing the process’ NSL and the FLT) and these obtained by the Hamiltonian- Lagrangian four-vector mechanical-phenomenological equation (including NSL and the complementary dynamical relationship), the so-called temporal evo- lution equation is obtained [10]. The temporal evolution equation relates the system’s Hamiltonian variations with entropy of the universe increasing, during a mechanical energy dissipation process, or Hamiltonian (mechanical energy) variations with Helmholtz free energy function (Gibbs free enthalpy function) decreasing in mechanical energy production processes. This paper is arranged as follows. In sec. 2 the system’s centre-of-inertia , and position, a relativistic NSL and its relativistic complementary dynamical relationship, are obtained; a relativistic Hamiltonian-Lagrangian formalism, with dissipation or production functions, is developed; and a four- vector mechanical-phenomenological equation is postulated, being applied to the block’s descending a friction incline process. In sec. 3, the developed for- malism is applied to a ball ascending on a frictionless incline powered by a chemical reaction. In sec. 4, evolution equations for these two processes are obtained by comparison of the four-vector mechanical and the four-vector fun- damental equation formalisms. In sec. 5 some conclusions are drawn on the subjects dealt with, and about its interest for students interested exploring deep relationships between mechanics and thermodynamics.

2 Block descending a friction incline

A relativistic Hamiltonian-Lagrangian formalism is going to be developed while applying it to block’s descending friction incline process. In the system’s centre-of-mass position and velocity definitions are quite simple; in relativity, due to Einstein’s inertia of energy principle, to describe the system’s

3 y yˆ

v i S v vˆj vj vˆi 0 ci xˆ j i O¯ rij ci rci rj ri ⌃ S

O x

Figure 1: Extended, composite system Σ (block), with s elements, a balls (with constant inertia) and b massless springs (no damped). Balls i and j distance rij. Reference frame S(x, y) with balls’ va. Zero linear momentum frame S0(ˆx, yˆ). Centre-of-inertia velocity vci and system’s characteristic coordinate rci. Ball a velocityv ˆa with respect to frame S0. characteristic point position and velocity (to whose variations NSL and com- plementary dynamical relationship equations are devoted) is somewhat more intricate. Consider a system Σ, a kind of block, formed by s elements, a solid bodies (balls, for example, assumed homogeneous and with constant inertia) and b springs. (Fig. 1). For an a element, starting with their constituent elementary particles, protons, neutrons and electrons, separated from each other, its inter- (a) nal energy E0 (ξ, V, T ), depending on their chemical composition ξ, volume V and temperature T , is given by [12]:

(a) 2 − ˜ E0 (ξ, V, T ) = Σnmnc U ξV + Nnc¯nT, (2) where mn is the mass of its n-th elementary particle, |U˜|ξ,V is its (negative) interaction internal energy, depending on the way elementary particles are or- ganized into nuclei, atoms and crystal, and on its volume (through external pressure), Nn is the number of mol andc ¯n its molar (mean) specific heat, with −2 (a) inertia Ma(ξ, V, T ) = c E0 (ξ, V, T ) [17]. Bodies in system Σ interact with each other by relativistic (massless) springs (no damped), able to store elastic energy Eb. One assumes that in frame S each element into the system can be characterized by its (instantaneous) position (ij) (xs, ys) and speed vs ≡ (vxs, vys). Energy E0 stored in a spring depends on rij, bodies i and j it joins distance. 2 In frame S body a has translational kinetic energy Ka = (γva − 1)Mac 2 2 −1/2 [where γv = (1 − v /c ) ]. The b ≡ ij spring (with elastic constant k (ij) 1 2 and length at rest x0) stores elastic potential energy E0 = 2 k(rij − x0) , −2 b having associated inertia Mb = c E0 . At a given instant, in frame S, the system’s linear momentum and energy are p = Σaγva Mava + Σbγvb Mbvb, and 2 2 E = Σaγva Mac + Σbγvb Mbc , respectivelly.

4 If each s element into the system can be identified by its inertia Ms, that will not change (its chemical composition, volume and temperature remain con- stant), and, at a given instant, by its geometrical centre position and velocity int ext vs, and forces, internal Fs and external Fs , exerted on each element are known, a Newton’s second law equation,

int ext Msd(γvs vs) = (Fs + Fs )dt , (3) can be proposed for each element in the system. For a system with many elements (typically, a > 2), the system of NSL equations cannot be solved. And when the internal forces are not known, this approach is non-operational. In classical physics the problem was solved by Newton by introducing New- ton’s third law, and writing equations that describe the behaviour, variations in velocity and position, of the system’s centre-of-mass. In relativity this concept is to be replaced by the concept of the centre- of-inertia [18, pp. 153-5]: because of Einstein’s inertia of energy principle, the system’s internal energy contributes to its inertia – and therefore to its linear momentum and its total energy. The composite system Σ centre-of-inertia speed vci in frame S is defined as [19, p. 246]

p Σaγva Mava + Σbγvb Mbvb vci = −2 = (4) c E Σaγva Ma + Σbγvb Mb

Velocity vci is the speed with which an observed in frame S [coordinates (x, v)], observes frame’s S0 [coordinates (ˆx, vˆ)] origin O¯ displacement. Frame S0 is (a) the system’s Σ its zero momentum frame, with p0 = Σaγvˆa Mavˆa = 0 and (b) p0 = Σbγvˆb Mbvˆb = 0, with centre-of-inertia relative speedv ˆs,v ˆs = (vs − 2 vci)/(1 − vsvci/c ). The system’s Σ inertia M (at a given instant) i.e., the system’s total energy in its zero momentum frame S0, is

M = Ma + Mb = Σaγvˆa Ma + Σbγvˆb Mb . (5)

2 From relationships γvˆs vˆs = γvs γvci (vs − vci), and γvˆs = γvs γvci (1 − vsvci/c ), for the a ball’s linear momentum pA in S one has pA = γvci (Σaγvˆa Ma)vci, because Σaγvˆa Mavˆa = 0, by definition, in S0; in a similar way, for energy EA 2 one has EA = γvci (Σaγvˆa Ma)c . Similar results are valid for the springs, with −2 −2 2 pB = γvci [Σbγvˆb c Eb]vci and EB = γvci [Σbγvˆb c Eb]c . Thus, for the whole system Σ in frame S,

p = pA + pB = γvci Mvci ≡ pci , (6) 2 E = EA + EB = γvci Mc ≡ Eci . (7)

Adding up on eqs. (3) for the s elements of the system Σ, Σsd(γvs Msvs) = ext int int Σs(Fs + Fs )dt, with internal forces sum zero, ΣsFs = 0, one has: ext Md(γvci vci) = F dt , (8)

ext ext where F ≡ ΣsFs . Eq. (8) is the relativistic NSL equation for a composite system whose inertia remains constant during a process and whose centre-of- inertia can be geometrically identified. For equation (8) to be operational, it

5 must be possible to choose a frame S in which forces are simultaneously applied during time interval [0, dt]. The characteristic system’s coordinate xcr is defined in frame S as [20]:

Σaγva Maxa + Σbγvb Mbxb xcr = , (9) Σaγva Ma + Σbγvb Mb where coordinates x are the coordinates of the geometric centers of the different elements of the system. In general, dxcr =6 vcidt, because displacement and velocity of each element do not correlate with those of the centre-of-inertia. When internal speeds arev ˆs = 0 (elements do not move with respect to each other), elements move with the same vci speed, and

Σaγvci Maxa + Σbγvci Mbxb ΣaMaxa xcr = ≈ ≡ xcm (10) Σaγvci Ma + Σbγvci Mb ΣaMa In this case (zero inertia springs), in frame S the centre-of-inertia coordinate coincides with classical centre-of-mass coordinate. It follows then that in that case dx cr = v → x ≡ x . (11) dt ci ci cr For (constant inertia M) system Σ, Newton’s second law eq. (14) complemen- tary dynamical relationship can be obtained, by multiplying it by vci, as

ext Mvcid(γvci vci) = F vcidt , (12) 2 ext Md(γvci c ) = F dxci , (13)

2 with dxci = vcidt; relationship vd(γvv) = d(γvc ) [9] has been used. Therefore, for a composite system, the operational equations, based on external forces and time intervals,

ext Md(γvci vci) = F dt , (14) 2 ext Md(γvci c ) = F dxci , (15) must refer to the variations in speed and position of the representative point of the system, usually its geometrical centre. Speed vci variation, given by eq. (14), and coordinate xci variation, given by eq. (15), characterize the sys- tem’s position and speed variations as a whole. These equations will not be operational if the inertia of the system varies during a process; they can be applied to describe finite processes when the system behaves as a particle with coordinates xci and vci. Relativistic Hamiltonian. For a block descending an incline with (dy- namic) friction coefficient µd, the frictional force does not perform work, i.e., it does not produce energy able to be stored in a work reservoir or in a mechanical potential; so it does not appear into the Hamiltonian. In frame S, with the incline at rest, coordinate x advances along the incline in the descending direction (Fig. 2). Block’s total mechanical energy is given by 2 2 2 1/2 2 E = (pcic + E0 ) = γvci Mc , (16)

6 0 y

fR N h µd x(t) G v(t) ↵ x Figure 2: Block descending an incline with friction. Frame S(x, y). A block, with initial velocity vi = 0, descends an incline, with block–incline coefficient of friction µd. When distance x(t) has been covered, in time interval [0, t], block’s linear speed is v(t).

2 where pci = γvci Mvci is block’s lineal momentum and E0 = Mc its internal (rest) energy. The block-Earth gravitational potential energy (with height hci given as hci = (L − xci) sin α, where L is incline’s length; when xci = L, gravitational potential energy is zero) is

MEM MEM Ep(xci) = −G + G ≈ Mg(L − xci) sin α , (17) RE + hci RE 2 (g = GME/RE) where xci is block’s displacement along the incline. Hamilto- nian for this system and process is [21]:

2 2 2 4 1/2 H(xci, pci) ≡ (pcic + M c ) + Mg (L − xci) sin α . (18) From first Hamilton’s equation dx ∂H(x , p ) p c2 ci ci ci ci → vci = = = 2 2 2 4 1/2 dt ∂pci (pcic + M c )

pci = γvci Mvci , (19) one recovers pci expression. The second Hamilton’s equation dp ∂H(x , p ) ci = − ci ci → (20) dt ∂xci

Md(γvci vci) = Mg sin α dt (21) does not provide the correct NSL for the process. Relativistic Lagrangian. From Hamiltonian (18) expressed as

2 H(xci, vci) ≡ (γvci − 1)Mc + Mg(L − xci) sin α , (22)

2 1 2 where K = (γvci − 1)Mc ≈ 2 Mvcm is block’s relativistic kinetic energy, 2 2 one has Lagrangian L(xci, vci) = vcipci − H(xci, vci) [22]– with [γv(v − c ) = −1 2 −γv c ]–[23]: − −1 M 2 − M − L(xci, vci) = (1 γvci ) c g(L xci) sin α , (23) ∗ − −1 M 2 ≈ 1 M 2 where K = (1 γvci ) c 2 vcm is block’s relativistic kinetic co-energy [24], and where xci is block’s centre-of-inertia position on the incline and vci its speed, increasing in the descending way.

7 From Lagrangian (23), the Euler-Lagrange equation is

d ∂L(xci, vci) ∂L(xci, vci) − = 0 → Md(γvci vci) = Mg sin α dt , (24) dt ∂vci ∂xci

−1 2 with d(−γv c )/dv = γvv [25]; it does not provide the correct NSL for the process. A Rayleigh’s dissipative function R(x) (phenomenological, based on Amon- tons-Coulomb frictional force [14]) must be introduced to take the frictional force into account [16]. In this case R(xci) = −µdMg cos α xci, with dR(xci)/dxci = −µdMg cos α. Hamilton’s second equation, with Rayleigh’s term is dp ∂H(x , p ) dR(x ) ci = − ci ci + ci → dt ∂xci dxci

Md(γvci vci) = M(sin α − µd cos α)g dt . (25)

Eq. (25) is the process’ accurate NSL. One has the Euler-Lagrange-Rayleigh equation [26],

d ∂L(x , v ) ∂L(x , v ) dR(x ) ci ci − ci ci = ci → dt ∂vci ∂xci dxci

Md(γvci vci) − Mg sin α dt = −µdMg cos α dt , (26) what is eq. (25). Multiplying eq. (26) both sides by vci, one has  Mvcid(γvci vci) = Mg sin α − µdMg cos α vcidt , (27) 2  Md(γvci c ) = Mg sin α − µdMg cos α dxci . (28)

Eq. (28) is NSL eq. (25) complementary dynamical relationship. Four-vector mechanical-phenomenological equation. For a point particle mass m, moving in frame S with velocity v = (vx, vy, vz), submitted to a conservative force F = (Fx,Fy,Fz), the four-vector impulse-linear momentum variation equation dpµ/dτ = F µ [19], four-vector momentum pµ and four- vector Minkowski’s force, F µ, can be expressed as

mdvµ = F µdτ , (29)     d(γvvx) γvFx  d(γvvy)   γvFy  −1 m   =   γv dt ,  d(γvvz)   γvFz  −1 d(γvc) c γvF · v

−1 with particle’s dτ = γv dt (t is frame’s S time). Previous considerations suggest that the four-vector space-time for a com- µ posite system is dxci = (dxci, dyci, dzci, cdt) and that its four-vector velocity is µ µ −1 vci = dxci/dτci = (γvci vxci, γvci vyci, γvci vzci, γvci c), where dτci = γvci dt. From eqs. (14) and (15), the four-vector mechanical equation eq. (29) is generalized for a composite body, inertia M, submitted to a resultant external force Fext. The postulated mechanical-phenomenological four-vector matrix

8 ext ext ext ext equation, with resultant external force, F = (Fx ,Fy ,Fz ), and system’s velocity vci = (vx, vy, vz) ≡ v in frame S, is

µ µ Mdvci = Fextdτci , (30)    ext  d(γvvx) Fx ext  d(γvvy)   Fy  −1 M   = γv  ext  γv dt ,  d(γvvz)   Fz  −1 ext d(γvc) c F · vci where dxci ≡ vcidt is the system’s centre-of-inercia vector displacement (term ext F · dxci contains, in general, pseudo-work as well as configuration work terms). Eq. (30) is valid in frame reference S in which forces are applied simultaneously during time interval [0, dt], where t is time displayed by the set µ µ µ µ of synchronized clocks in S [27]. With cdp = dKci and cFextdt = δpWext, one µ µ has (constant inertia, dE = dKci)

µ µ dKci = δpWext , (31)    ext  cMd(γvvx) cFx dt ext  cMd(γvvy)   cFy dt    =  ext  , (32)  cMd(γvvz)   cFz dt  2 ext Md(γvc ) F · dxci where pWext is for pseudo-work. Setting up four-vector equations presents advantage that when a process is going to be described in frame S,¯ in standard configuration with respect to frame S, with speed V , equations describing the process in S¯ are obtained µ by applying the Lorentz transformation Lν (V ) to the four-vector equation in µ ν ν S ( asynchrous formulation), with Lν (V )[dKci = δpWext] → ¯ µ ¯ µ dKci = δpWext, preserving the principle of relativity [9]. Descending block four-vector mechanical-phenomenological equa- tion. For block descending under gravity on a friction incline process, with N = Mg cos α, one has

[G + N + fR] · dxci ≡ ,

≡ [(Mg sin α, −Mg cos α, 0) + (0,N, 0) + (−µdN, 0, 0)] · (dxci, 0, 0) =

= Mg(sin α − µd cos α)dxci . (33)

By integrating the four-vector differential equation during time interval [0, t0], with vi = 0, one has     cγv0 Mv0 cMg(sin α − µd cos α) t0  0   c(−Mg cos α + N) t0    =   , (34)  0   0  2 (γv0 − 1)Mc Mg(sin α − µd cos α) x0 where v0 ≡ vci(t0) and x0 ≡ xci(t0) are block’s centre-of-inertia speed and dis- placement at time t0. The NSL and its complementary dynamical relationship

9 for the process are (∆E = ∆K):

∆p ≡ Mγv(t0)v0 = Mg(sin α − µd cos α) t0 , (35) 2 ∆K ≡ M(γv(t0) − 1)c = Mg(sin α − µd cos α) x0 , (36) Eq. (34) integrates NSL and its complementary dynamical relationship for the process. For the sake of comparison, the four-vector fundamental equation for µ µ µ µ the process Ef − Ei = ΣkWk + Q , is [12]:         cγv0 Mv0 0 cMg(sin α − µd cos α) t0 0  0   0   c(−Mg cos α + N) t0   0    −   =   +   ,  0   0   0   0  2 2 γv0 Mc Mc Mg sin α x0 Q(t0) (37) where Q(t0) is mechanical energy dissipated as heat at time t0. Eq. (37) inte- grates NSL and the FLT for the process.

y v(t) x(t) N Fa/b h x F⇠/b 0 Fb/⇠ G


Figure 3: Cannonball, with inertia M, initial velocity vi = 0, ascends an incline angle α, powered by a chemical reaction taking place in a cannon. Frame S(x, y). Frame S0 is located at the centre of the ball. The cannon exerts force Fc/ξ on chemicals and chemicals exert force Fξ/b on cannonball. When the ball has covered distance x(t), in time interval [0, t], its velocity is v(t) and its height is h(t). The x-axis advances up the incline.

3 Cannonball rising on incline

For a cannonball ascending a frictionless incline by force Fξ/b exerted on it by products of a chemical reaction (Fig. 3), it is not possible to find a mechan- ical potential Vξ(x) such that Fξ/b = −dVξ(x)/dx. Force Fξ/b arrives from a potential variation, not from a mechanical one but from a thermodynamic potential, Helmholtz’s free energy function ∆Fξ (maximum work), or Gibbs’ free enthalpy function ∆Gξ (maximum mechanical energy) variation, for the chemical reaction. A thermodynamic potential, and force provided by a chem- ical reaction, cannot be directly incorporated into the Hamiltonian-Lagrangian formalism. Atmosphere, pressure P ext, is considered a work reservoir - it can perform

10 positive or negative work on the system –, which stores a potential energy ext Va(x) = P Ax when cannonball position is x; potential Va(x) is incorporated into the Hamiltonian. The cannonball-rising-incline process Hamiltonian is given as (in what fol- lows, x ≡ xci and v ≡ vci):

H(x, p) ≡ (p2c2 + M2c4)1/2 + Mgx sin α + P extAx , (38) with Adx = dVξ. Ball’s internal energy, and its inertia, remain constant. At- mosphere exerts force Fa/b = −dVa(x)/dx on the ball. Ball’s potential energy increases in incline’s upward way (x increases upwards). From Hamiltonian (38) Hamilton’s second equation does not provide the correct NSL for the pro- cess. With Hamiltonian expressed as

2 ext H(x, v) ≡ (γv − 1)Mc + Mgx sin α + P Ax , (39) the Lagrangian for the ball-ascending-incline process is

−1 2 ext L(x, v) = (1 − γv )Mc − Mgx sin α − P Ax . (40)

From Lagrangian (40) Euler-Lagrange equation, one does not obtain the correct NSL for the ascending cannonball. In order to take into account force Fξ/b resulting from chemical reactions (i.e., not given as a mechanical potential gradient), a mechanical energy pro- duction function is introduced. For example, assuming a constant force, one can consider a Gibbs’ mechanical energy production function G(x) = Fξ/bx, with dG(x)/dx = Fξ/b. The Euler-Lagrange-Gibbs equation, for ball’s translation ascending the incline by consuming fuel, is

d ∂L(x, v) ∂L(x, v) dG(x) − = . (41) dt ∂v ∂x dx Applying eq. (41) to Lagrangian (40) and function G(x), the accurate impulse- linear momentum variation equation

ext Md(γvv) = [Fξ/b − (Mg sin α) + P A)]dt , (42) is obtained; its complementary dynamical relationship (pseudo-work) is

2 ext Md[(γv − 1)c ] = [Fξ/b − (Mg sin α) + P A)]dx . (43)

Four-vector mechanic equation. Same results are directly obtained µ µ from the four-vector mechanical equation ∆Kci = pW (system cannonball),

    ext   cγv0 Mv0 c Fξ/b − (Mg sin α + P A) t0  0   c(−Mg cos α + N) t0    =   , (44)  0   0  2  ext  (γv0 − 1)Mc Fξ/b − (Mg sin α + P A) x0

11 where v0 ≡ v(t0) and x0 ≡ x0 are projectile’s centre-of-inertia speed and dis- placement at time t0, respectively. By integrating the above expressions during time interval [0, t0], NSL and complementary dynamical relatioship equations are

ext ∆p ≡ γv0 Mv0 = [Fξ/b − (Mg sin α + P A)] t0 , (45) 2 ext ∆K ≡ (γv0 − 1)Mc = [Fξ/b − (Mg sin α + P A)] x0 . (46)

The four-vector fundamental equation for the process (system cannonball- µ µ µ µ chemical reaction) Ef − Ei = ΣkWk + ∆Gξ is [12]:         cγv0 Mv0 0 c[Fξ/b − (Mg sin α + PA)]t0 0  0   0   −c(Mg cos α − N)t0   0   −  =  +  ,  0   0   0   0  2 2 γv0 Mc Mc −Mgx0 sin α −n0∆gξ (47) where n0 ≡ n(t0) is mol of reactants consumed at time t0. For Helmholtz’s free energy function, with ∆Fξ = ∆Gξ − P ∆Vξ, one has the FLT equation:

2 −∆Fξ = (γv0 − 1)Mc + Mgx0 sin α + P ∆Vξ . (48)

4 Temporal evolution equations

For these processes, one has acceleration parameters aR and aξ, with kinematic relationships:

aR = (sin α − µd cos α)g , (49)

γvv(t) = aRt , (50) 2 (γv − 1)c = aRx(t) , (51) −1 ext aξ = M (Fξ/b − P A) − g sin α , (52)

γvv(t) = aξt , (53) 2 (γv − 1)c = aξx(t) . (54)

The mechanical description of these two processes does not discriminate be- tween force fR and force Fξ/b. One must appeal to processes’ thermodynamic description to better understanding different role played by them.

4.1 Block moving on friction incline

Mechanical energy temporal evolution. With the expression of Hamil- tonian (corresponding to block’s energy in its gravitational interaction with 2 Earth) H(x, v) = (γv − 1)Mc + Mg (L − x) sin α, and the complementary  2  dynamical relationship eq. (36) expressed as d M(γv − 1)c − Mgx sin α = −d[µdMgx cos α], one obtains dH(x, v) dR(x) = = −f v < 0 . (55) dt dt R

12 Mechanical energy dissipates in this process. Dissipated mechanical energy, given by Wd(t) = −µdMg cos α x(t), is quantified as a force, the friction force, times block’s centre-of-inertia displacement scalar product, a pseudo- work quantity [28]. Lagrangian eq. (23) is time t symmetric, it is time independent [∂L(x, v)/∂t = 0]. However, mechanical energy is not conserved throughout the process (Noether’s theorem conditions for process’ temporal inversion symmetry are not fulfilled [13, pp. 236-9], because the non-conservative frictional force), with

2 d[(γv − 1)Mc − Mgx sin α] = −µdMg cos α v < 0 . (56)

Hamilton’s principle (principle of stationary action) is not satisfied for this process [13, p. 227]. Entropy of the universe temporal evolution. The results achieved using the four-vector fundamental equation and them from the Hamiltonian- Lagrangian formalism, can be compared. It has been previously obtained [12] that this process evolves with entropy of the universe increment [29],

dS T U = f v > 0 . (57) dt R By comparing eq. (55) with eq. (57), one has [10]

dH(x, v) dS = −T U . (58) dt dt By trying to attain gravitational potential energy minimum (to increase the entropy of the universe at maximum), the block dissipates mechanical energy, with the frictional force as the mechanism to do it. Eq. (58) relates dissipated energy mechanical description, in terms of Hamil- tonian decreasing quantity and dissipative (negative) work done by the block on the incline, with the thermodynamics description of (the same amount of) energy dissipated in terms of heat exchanged and increased entropy of the uni- verse. Of course, for µd = 0, equations describe a frictionless process, evolving with mechanical energy conservation. Eq. (58) is valid in classical and rela- tivistic physics. Potential energy minimum principle. Let us suppose that a system subjected to a dissipative force evolves between two states characterized by body’s position and speed, state 1, with (x1, v1) and state 2, with (x2, v2). The body begins and ends with equal kinetic energy, v2 = v1, such that in the process dK = 0. Temporal evolution equation (58) is then

dV (x) dS = −T U , (59) dt dt where dV (x) is potential energy variation between states 1 and 2. Eq. (59) mathematically expresses the so-called principle of potential energy minimum [30, Ch. 4] (maximum entropy law): whenever a frictional force intervenes as much mechanical energy will be dissipated as possible. This can be seen better

13 when the movement is periodic, with v2 = v1 = 0. When a damped pendulum swings, mechanical energy dissipates when it is moving. When it reaches rest, it has less and less amplitude or height. To some extent, eq. (59) could be considered the mathematical-physical description of Murphy’s law [31].

4.2 Projectile launched by a chemical reaction Helmholtz’s free energy function temporal evolution. For a thermo- dynamic process, with internal energy variation ∆U, for example, and entropy variation ∆S < 0, the maximum work is obtained from internal energy decreas- ing subtracting minimum heat Qmn = T ∆S released to the thermal reservoir to ensure that the entropy-of-the-universe does not decrease during the process. The maximum work available from that process, including mechanical energy produced and work expansion against the atmosphere, is Helmholtz free energy function, ∆F = ∆U − T ∆S, decreasing during the process, with W mx = −∆F (in general, the obtained work W is W ≤ W mx). From eq. (48) one has for this process

 2  d (γv − 1)Mc + Mgx sin α + PVξ ≈ −dn∆fξ , (60) where n∆fξ = ∆Fξ, with ∆Fξ = ∆Uξ −T ∆Sξ is chemical reaction Helmholtz’s free energy function variation. By comparing the FLT eq. (60) and comple- mentary dynamical relationship (43), Hamiltonian (39) temporal variation is related to Helmholtz free energy decreasing as dH(x, v) dF = − ξ = F v > 0 . (61) dt dt ξ/b

For the system evolving between two states, 1, with (x1, v1) and 2, with (x2, v2), such that v1 = v2 = 0 (a vertically launched projectile starts with vi = 0 and when it reaches its maximum height hmx it has v2 = 0), it is dV (x) dF = − ξ > 0 . (62) dt dt For a thermal engine (working with a gas or with water steam, for example) performing a Carnot’s cycle (W = W mx), consuming fuel, for example, by burning coal, eq. (62) summarizes its operation surrounded by an external atmosphere. The gas performing the cycle does not vary its thermodynamic quantities, ∆V (x) ≤ W mx is work obtained by a cycle (work by raising an amount of water against the Earth’s gravitational field, plus expansion work against external atmospheric pressure, using a Watt engine, for example) and −∆Fξ ≥ W is the decrease in Helmholtz function for fuel burned to the thermal engine. Gibb’s free enthalpy function temporal evolution. When work expansion against the atmosphere, WP = −P ∆Vξ, is removed from maximum work available by decreasing Helmholtz free energy function, the maximum available mechanical energy, is equal to Gibbs’ free enthalpy function, ∆Gξ = mx ∆Uξ + P ∆Vξ − T ∆Sξ, decreasing during the process, with Em = −∆Gξ (in

14 mx general, obtained mechanical energy Em is Em ≤ Em ). For this system and process the mechanical energy (excluding the atmospheric pressure work) is 2 Em(x, v) = (γv − 1)Mc + Mg x sin α. The FLT equation for the whole system 2 projectile plus chemicals is expressed as d[(γv − 1)Mc + Mgx sin α] = −dGξ, where dGξ is Gibbs’ free enthalpy function variation during the process. By comparing these equations, one gets [10]

dE (x, v) dG m = − ξ > 0 . (63) dt dt This equation relates the rate of mechanical energy production to the rate at which Gibbs’ function decreases throughout the process [11]. When a thermodynamic cycle is analyzed, mechanical energy Em obtained (rising water gravitational potential energy, for example) is less, or equal, than fuel consumed Gibbs’ function decreasing, Em ≤ −∆Gξ. Eq. (63) would be the expression of (no-Murphy’s law [32] or) Gybbs’ law.

5 Conclusions

The aim of this work is to extend the equations describing centre-of-mass velocity and position variations of a composite body, submitted to conservative and non-conservative forces, to relativity, using the methods and concepts of Einstein’s special relativity, integrating them into a relativistic Hamiltonian-Lagrangian formalism. The system’s centre-of-inertia velocity and position concepts, have been developed, and relativistic Newton’s second law and complementary dynami- cal relationship equations, describing these quantities evolution throughout a process, have been proposed. A Hamiltonian-Lagrangian formalism has been generalized to deal in rela- tivity with mechanical energy dissipation, or with mechanical energy produc- tion, processes; this formalism is expressed in a four-vector matrix equation, incorporating Newton’s second law and its complementary dynamical relation- ship in a single expresion. For a process involving a dissipative force, block on friction incline, the Euler-Lagrange equation must be generalized by introducing a Rayleigh’s dis- sipation function. This function allows obtaining accurate Newton’s second law equation and then its complementary dynamical relationship. By comparison to results previously obtained with the four-vector fundamental equation, a temporal evolution equation is obtained relating system’s mechanical energy decreasing with entropy of the universe increasing. For a cannonball launched by a chemical reaction, producing mechanical en- ergy, a Gibbs’ production function is considered into the Euler-Lagrange equa- tion, which allows obtaining the correct equation for Newton’s second law. By comparison with the results achieved using the four-vector fundamental equa- tion, the temporal evolution equation for the mechanical energy is obtained, showing that produced mechanical energy comes from Gibbs’s free enthalpy function decreasing for chemical reaction.

15 Topics covered here can be studied at the end of a sophomore physics year, when students have courses in mechanics, including Hamiltonian-Lagrangian formalism and special relativity – at least an introduction –, as well as a course in thermodynamics. Since thermodynamics is ultimately a study of complex systems, it is not by chance that students see it as a difficult discipline. An advanced undergraduate student will appreciate that the laws of mechanics and thermodynamics are presented jointly by using Einstein’s relativity in four- vector formalisms. The special theory of relativity is more than the correct theory for processes occurring at speeds close the ; it is the conceptual framework (including the Lorentz transformation) within which problems can, and must, be solved. A four-vector formalism physics and mathematics requirements make it possible to give a precise physics meaning to quantities involved, which are part of a four-vector, and to obtained equations. These formalisms avoid confusion (between the complementary dynamical relationship and the first law of thermodynamics, for example) and provide physics insights.


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