A variational formulation for dissipative systems

International/Interdisciplinary Seminar on Nonlinear Science The University of Tokyo, komaba Campus March 22, 2017 ------------------------------------------------------------------------------- A variational formulation for dissipative systems Hiroki Fukagawa Faculty of Engineering, Kyushu University 1

TOPICS 1 In terms of an optimized control theory, the realized dynamics of physical systems take stationary values of cost functionals. 2 The equations of motion for dissipative systems are obtained by solving stationary problems subject to non-holonomic constraints for entropy. 3 The non-holonomic constraints are determined to satisfy the law of entropy increase, symmetries, and well-posedness. 2

How to describe physics The time development of a physical system can be described as a curve C in configuration space q and time t. q t 4

Variational principles A realized motion gives a stationary value of a functional. C L(q, q,t ) dt Euler-Lagrange Eq. q q L d L =0 q dt q t : time 5

The dynamics of a harmonic oscillator 1 2 1 2 Action functional C 2 q 2 q dt ( q ) Euler-Lagrange Eq. n o i t i s o p : q t : tim e q +q=0 ()( ) q = cos t q sin t 6

Variational calculus of an action functional u q t : tim e 7

dq The relation: u=0 yields dt 8

+) where p= L / u Euler-Lagrange eq. Energy conservation 9

Method of Lagrange multiplier = ~ H ( p, q) H ( p, q, u ( p, q)) ~ H where u* (p,q) is the solution of =0. u Hamilton s eq. 10

A harmonic oscillator 1 2 1 2 L(q, u)= u q, 2 2 ~ H ( p, q, u)= pu L(q, u) ~ Put u ( p, q)= H = p u 1 2 1 2 H ( p, q)= p + q 2 2 u Hamilton s equations dq H = = p=cos t dt p dp H = = q= sin t dt q q t : tim e 11

Pontryagin s maximum principle (PMP) Find an optimized control u* which gives the stationary value of a cost functional,. where ~ H ( p, q, u)= pf (q, u) L (q, u) Hamilton s equations dq H = dt p where dp H = dt q u q t ~ H ( p, q) H ( p, q, u ( p, q)) ~ u* (p,q) is the solution of H =0. u 12

A dissipative system C dq u dt=0 0 q:position t:time 14

A dissipative system C 0 ds+ ζ dq+ J dt=0 s:entropy t:time 15

A dissipative system C s:entropy 0 ds+ ζ dq+ J dt=0 s:entropy q:position q:position t:time t:time 16

A holonomic constraint C 0 ds+ ζ dq+ J dt=0 C 0 Where Cα s C0 q t 17

A holonomic constraint where Cα s C0 q t 18

where Cα s C0 q t 19

+) where,, and. 20

Method of Lagrange multiplier ~ H ( p, q, s) H ( p, q, s, u ( p, q, s)) Equations of motion ~ where u* (p,q,s) is the solution of H =0. u s g (q, t )=0 s q t 21

A holonomic constraint where Cα s C0 q 22

A dissipative system (A non holonomic constraint) s:entropy q:position s:entropy q:position t:time No surface on which the curves lie. 23

ds+ζdq+jdt is orthogonal to the tangent vector of C0 and the virtual displacement X = α(δt,δq,δs). ds+ζdq+jdt s Cα C0 q t 24

+) where,, and. 25

A damped oscillator s:entropy q:position t:time ( 1 2 1 2 ~ H ( p, q, u)= pu u q 2 2 1 1 H ( p, q)= p2 + q 2 2 2 Equations of motion: dq =p dt dp = q f dt 26 )

Entropy production ds dq = ζ J dt dt >0 Thus ζ is opposite sign of dq/dt. s:entropy dq/dt q:position ζ s J t:time 29

Damped Oscillators k m1 s m2 S Lagrangian Constraint 30

Damped Oscillators k m1 s m2 S = m i ui = p i = dq i = pi dt = dp i = k (q 1 q 2 )+ f i dt 31

Space translation symmetry qi Y=α(gq,gs,gt) t The sufficient condition of LY (Tds+ f i dq i +Q dt )=0 where LY =(d ι Y +ι Y d ) is Thus, 32

Evaporation nt te la at he e c a on f r si u S en t Liquid Gas 34

Evaporation Surface energy ρ Mass density ρ is a function of the initial position q. 質量密度 ρ The Lagrange derivative of the initial position q is zero. radius Then we have 35

Evaporation Surface energy ρ 質量密度 ρ radius 36

Taking the variation of where - yields δ = The surface term imposes an extra boundary condition. + 37

Well-posedness Mathematical models of physical phenomena should have the properties that (1) A solution exists (2) The solution is unique (3) The solution depends continuously on the initial conditions and the boundary conditions. In order to have a solution, eliminate the extra surface term by the non-holonomic constraint σjieij where i ik e j δ ( j u k + k u j ) 38

Equations of motion j j j t ( ρ u i )+ j ( ρ u u i + Π i + σ i ) γ i =0 ( ) 1 E E E j Π P ρ T k +ρ E δ i + i ρ ρ ρ ρ T k j j i 39

Summary The dynamics of a physical system can be described by a trajectory in a configuration space. The trajectory is determined by a law of its tangent vector, i.e., the generalized velocity. In terms of a control theory, the velocity is regarded as the control parameter which determines the trajectory. The law of velocity is universalized into a policy of control. Hamilton s principle is generalized as an optimal control theory, which seeks the control giving the stationary value of a cost functional. In a dissipative system, entropy depends on the time development of other variables in the configuration space. This relation is given by a set of non-holonomic constraints. Then the equations of motion are obtained by solving the stationary condition of a cost functional subject to the nonholonomic constraints. In this formulation, physical systems are characterized by the sets of a functional and non-holonomic constraints. All are consistent with symmetries and well-posedness. Moreover, the constraint of entropy satisfies the law of entropy increase. These restrictions define the proper class for equations of motion in physics. 40

References H. Fukagawa and Y. Fujitani: Prog. Theor. Phys. 124(2010)517; ibid.127(2012)921. H. Fukagawa, C. Liu and T. Tsuji: arxiv: 1411.6760(2014). H. Fukagawa: A Variational Principle for Dissipative Systems, Butsuri 72-1 (2017)34. (in Japanese) T. Kambe: Fluid Dyn. Res. 39(2007)98; ibid. 40(2008)399; Physica D 237(2008)2067. Z. Yoshida: J. Math. Phys. 50(2009)113101. B. Eisenberg, Y. Hyon and C. Liu: J. Chem. Phys. 133(2010)104104. L. S. Pontryagin: Mathematical theory of optimal processes CRC Press 1987. A. Onuki: Phys. Rev. E 76(2007)061126. 41