Relativistic mechanics and thermodynamics. II. A linear translation Hamiltonian-Lagrangian formalism J G¨u´emez Applied Physics 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 forces is developed. A block descending an incline with a frictional force, mechanical energy 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, mass 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 acceleration, 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 momentum 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 kinetic energy 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 energies) equation is essentially a linear momentum equation [3], including, in general, work (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 theory of relativity, including the inertia 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 position, 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 velocity, 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 classical physics 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' velocities va. Zero linear momentum frame S0(^x; y^). Centre-of-inertia velocity vci and system's characteristic coordinate rci. Ball a velocityv ^a 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