Abstract
The authors propose a formulation of the least action principle in dissipative systems through Lagrange equations.
Highlights
Despite the warning of the French mathematician Henri Poincaré, who stated in 1908 that “irreversible phenomena cannot be explained by means of Lagrange equations” [1], it has been the dream of many physicists to reconcile the thermodynamics of irreversible processes with the least action principle
In Eq (19), the generalized stiffness matrix T = [Tk j] as defined by Tisza is introduced [23]. This matrix is necessarily (i) symmetric while the equilibrium potential seq(e) satisfies the Maxwell relations and (ii) positive definite as long as the equilibrium around which the system is considered is stable. It is commonly accepted in the thermodynamics of irreversible processes that the nonequilibrium forces Xα and the nonequilibrium flows jα are linked via a coupling matrix L = [Lαβ ]
We have shown that any thermodynamic potential satisfies a differential conservation law which, once generalized in the sense of extended irreversible thermodynamics and integrated over a finite volume, fulfills an incremental least action principle in time
Summary
Despite the warning of the French mathematician Henri Poincaré, who stated in 1908 that “irreversible phenomena cannot be explained by means of Lagrange equations” [1], it has been the dream of many physicists to reconcile the thermodynamics of irreversible processes with the least action principle. The first one consists of multiplying the initial system (P) by a nondegenerate matrix such that the new system is self-adjoint An example of this strategy has been proposed in [7] for the oscillator previously presented [see Table I, irreversible case (a)], and in thermics for the Cattanéo-Vernotte equation and for an electrical circuit containing resistors [8]. We have presented the Rayleigh function of our oscillator, as irreversible case (c) This strategy has been used in continuum mechanics [13,14,15,16], and has an incremental formulation in terms of differential geometry [17]. A detailed example corresponding to a simple physical process (heat conduction) is considered to illustrate the approach, clarify technical aspects, and discuss the link of the present approach with the extended irreversible thermodynamics [19]
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