PHYSICAL REVIEW E 91, 013203 (2015)

Rayleigh-Lagrange formalism for classical dissipative systems Epifanio G. Virga* Dipartimento di Matematica, Universit`a di Pavia, Via Ferrata 5, I-27100 Pavia, Italy (Received 26 June 2014; revised manuscript received 24 September 2014; published 14 January 2015) It is often believed that the Rayleigh-Lagrange formalism for classical dissipative systems is unable to encompass forces described by nonlinear functions of the velocities. Here we show that this is indeed a misconception. DOI: 10.1103/PhysRevE.91.013203

PACS number(s): 45.20.−d, 45.50.−j

Dissipative systems are ubiquitous in physics but their description in general mathematical terms is still a debated issue. Galley [1] proposed a general approach to the motion of a discrete nonconservative system based on a modified Hamilton principle, which remedies the time-reversibility of the Lagrange equations, derived from the stationarity of the classical action. Dissipative systems are special nonconservative systems, and so Galley’s general method also applies to them. One motivation for developing this method was the supposed inability of the classical Rayleigh-Lagrange equations to account for resistive forces other than those linear in the velocities. It is my intention to show that a (slight) extension of the classical Rayleigh-Lagrange formalism is able to encompass general dissipative potentials, also amenable to a variational formulation. The notation employed here for a discrete dynamical system is standard. The generalized coordinates form the vector q = (q1 , . . . ,qm ) ∈ Rm ; correspondingly, q˙ = (q˙1 , . . . ,q˙m ) is ˙ is the kinetic the vector of generalized velocities. T (q,q) ˙ energy, taken to be a quadratic positive definite form in q; V (q) is the potential energy of all conservative generalized ˙ := T − V is the forces Q, so that Q = − ∂V , and L(q,q) ∂q Lagrangian of the system, which enjoys the usual smoothness ˙ := T + V be the total energy, it assumptions. Letting H (q,q) is an immediate consequence of T being quadratic in q˙ that H˙ + W = 0, where    m ∂L δL  d ∂L = q˙j W := q˙ · − δq ∂qj dt ∂ q˙j j =1   δT (1) = q˙ · Q + δq is the total mechanical power expended by active (Q) and inertial ( δT ) forces [2, p. 115]. Were these the only forces at δq work, standard Lagrange equations would require δL to vanish δq identically along any motion, and so would W , implying that the total energy is conserved. We assume that nonconservative generalized forces can be ˙ is a dissipation potential, not , where R(q,q) expressed as − ∂R ∂ q˙ ˙ Generalized Rayleigh-Lagrange necessarily quadratic in q. equations can then be written in the traditional textbook form, δL − ∂R = 0, which combined with Eq. (1) transform the δq ∂ q˙

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balance of energy into ∂R H˙ = −q˙ · =: −D, ∂ q˙

(2)

˙ is to be interpreted as the dissipation in the where D(q,q) system. Here we want that D be the only constitutive function ˙ but for the dissipative forces: it is positive semidefinite in q, not necessarily quadratic, and it vanishes identically whenever q˙ = 0. The dissipation potential R is determined in terms of D as a solution to the partial differential equation (2)2 . By applying the method of characteristics to this equation [3, Ch. II], we easily arrive at the following explicit representation for R (to within an arbitrary constant):    s ˙  . ˙ = (3) R(q,q) D(q,e q)ds s=0

˙ then so is R. If It follows from Eq. (3) that if D is convex in q, D is a homogeneous function of degree n, then R = n1 D.1 If ˙ then with Dn homogeneous of degree n in q, D= N n=1 Dn , N 1 by Eq. (3) R = n=1 n Dn . The generalized Rayleigh-Lagrange equations also admit a variational formulation in terms of the principle of reduced dissipation [2, p. 121], which states that for a discrete system with total mechanical power W as in Eq. (1) and dissipation potential R that obeys Eq. (2)2 the true velocity q˙ traversing a given configuration q is such that the reduced dissipation

:= R − W is stationary with respect to all virtual potential R are held fixed. variations δ q˙ once the generalized forces δL δq In conclusion, a generalized dissipation potential derived from a (not necessarily quadratic) dissipation function can be set as the basis of the Rayleigh-Lagrange dynamics of a discrete system. This shows that nonlinear dissipative forces can effectively be treated in a variational formalism within the Rayleigh-Lagrange framework. This is not the only proposal for a formulation of dissipative systems attempting to overcome the inability of classical variational principles to encompass them. Special note should be taken of Gurtin’s [5,6] and Tonti’s [7] principles of stationarity for convolved action functionals, recently brought in connection by Dargush and Kim [8,9] with a different stream of thought that regards fractional calculus as the appropriate mathematical tool to describe dissipation [10,11]. Compared to these, the view taken in this note is rather minimalist and more in tune with d’Alembert’s than Hamilton’s principle. Its merits are perhaps

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An instance with n = 3 is discussed for example in Ref. [4]. ©2015 American Physical Society

EPIFANIO G. VIRGA

PHYSICAL REVIEW E 91, 013203 (2015)

more apparent for continuum systems, possibly traversed by singularities, whose dynamical equations are not known in advance, but for which appropriate forms for the dissipation functional are more likely to be imagined on physical and symmetry grounds [4]. However, heeding that the classical Rayleigh-Lagrange equations have indeed a wider scope may be equally significant.

[1] C. R. Galley, Classical mechanics of nonconservative systems, Phys. Rev. Lett. 110, 174301 (2013). [2] A. M. Sonnet and E. G. Virga, Dissipative Ordered Fluids: Theories for Liquid Crystals (Springer, New York, 2012). [3] R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley, New York, 1962), Vol. II: Partial Differential Equations. [4] E. G. Virga, Chain paradoxes, Proc. R. Soc. London A 471, 20140657 (2014). [5] M. E. Gurtin, Variational principles for linear initial value problems, Quart. Appl. Math. 22, 252 (1964). [6] M. E. Gurtin, Variational principles for linear elastodynamics, Arch. Rational Mech. Anal. 16, 34 (1964).

ACKNOWLEDGMENTS

I am indebted to James A. Hanna, who in the course of a stimulating, ongoing correspondence suggested reading Ref. [1] and to an anonymous referee who drew my attention to a body of literature, part of which very recent, concerned with convolved action principles.

[7] E. Tonti, On the variational formulation for linear initial value problems, Ann. Mat. Pura Appl. 95, 331 (1973). [8] G. F. Dargush and J. Kim, Mixed convolved action, Phys. Rev. E 85, 066606 (2012). [9] G. F. Dargush, Mixed convolved action for classical and fractional-derivative dissipative dynamical systems, Phys. Rev. E 86, 066606 (2012). [10] F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E 53, 1890 (1996). [11] F. Riewe, Mechanics with fractional derivatives, Phys. Rev. E 55, 3581 (1997).

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Rayleigh-Lagrange formalism for classical dissipative systems.

It is often believed that the Rayleigh-Lagrange formalism for classical dissipative systems is unable to encompass forces described by nonlinear funct...
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