12.006J F2022 Lectures 6–7: Dissipation and Contraction of Volumes in Phase Space

Lecture notes for 12.006J/18.353J/2.050J, Nonlinear Dynamics: Chaos D. H. Rothman, MIT September 16, 2022 Contents 1 Conservation of volume in phase space 1 2 Damped oscillators and dissipative systems 2.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Phase portrait of damped pendulum . . . . . . . . . . . . . . 2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5 6 9 1 Conservation of volume in phase space Reference: Tolman [1] We show (via the example of the pendulum) that frictionless systems conserve volumes (or areas) in phase space. Conversely, we shall see, dissipative systems contract volumes. Suppose we have a 3-D phase space, such that x˙1 = f1 (x1 , x2 , x3 ) x˙2 = f2 (x1 , x2 , x3 ) x˙3 = f3 (x1 , x2 , x3 ) or d~x = f~(~x) dt The equations describe a flow, where d~x/dt is the velocity. A set of initial conditions enclosed in a volume V flows to another position 1 in phase space, where it occupies a volume V 0 , neither necessarily the same shape nor size: x3 n V’ x V x2 ds x1 Assume the volume V has surface S. Let • ρ = density of initial conditions in V ; • ρf~ = rate of flow of points (trajectories emanating from initial conditions) through unit area perpendicular to the direction of flow; • ds = a small region of S; and • ~n = the unit normal (outward) to ds. Then Z net flux of points out of S = or (ρf~ · ~n)ds S Z Z ∂ρ dV = − (ρf~ · ~n)ds ∂t S V i.e., a positive flux =⇒ a loss of “mass.” Now we apply the divergence theorem to convert the integral of the vector field ρf~ on the surface S to a volume integral: Z Z ∂ρ ~ · (ρf~)]dV dV = − [∇ V ∂t V Letting the volume V shrink, we have ∂ρ ~ · (ρf~) = −∇ ∂t 2 Now follow the motion of V to V 0 in time δt: V’ V The boundary deforms, but it always contains the same points. We wish to calculate dρ/dt, which is the rate of change of ρ as the volume moves: dρ ∂ρ ∂ρ dx1 ∂ρ dx2 ∂ρ dx3 = + + + dt ∂t ∂x1 dt ∂x2 dt ∂x3 dt ~ · (ρf~) + (∇ρ) ~ · f~ = −∇ ~ · f~ − ρ∇ ~ · f~ + (∇ρ) ~ · f~ = −(∇ρ) ~ · f~. = −ρ∇ Note that the number of points in V is N = ρV. Since points are neither created nor destroyed we must have dN dρ dV = V +ρ = 0. dt dt dt Thus, by our previous result, ~ · f~ = −ρ dV −ρV ∇ dt or 1 dV ~ · f~ =∇ V dt This is called the Lie derivative. We shall refer to it often in this class. We next arrive at the following statements by example: 3 ~ · f~ = 0 ⇒ volumes in phase space are conserved. Characteristic of • ∇ conservative or Hamiltonian systems. ~ · f~ < 0 ⇒ dV /dt < 0 ⇒ volumes in phase space contract. Character• ∇ istic of dissipative systems. We use the example of the pendulum: x˙1 = f1 (x1 , x2 ) = x2 g x˙2 = f2 (x1 , x2 ) = − sin x1 l Calculate ~ · f~ = ∂ x˙1 + ∂ x˙2 = 0 + 0 ∇ ∂x1 ∂x2 Pictorially x2 x1 Note that the area is conserved. Conservation of areas holds for all conserved systems. This is conventionally derived from Hamiltonian mechanics and the canonical form of equations of motion. In conservative systems, the conservation of volumes in phase space is known as Liouville’s theorem. 2 Damped oscillators and dissipative systems References: Bergé et al. [2], Strogatz [3] 4 2.1 General remarks We have seen how conservative systems behave in phase space. What about dissipative systems? What is a fundamental difference between dissipative systems and conservative systems, aside from volume contraction and energy dissipation? • Conservative systems are invariant under time reversal. • Dissipative systems are not; they are irreversible. Consider again the undamped pendulum: d2 θ + ω 2 sin θ = 0. 2 dt Let t → −t and thus ∂/∂t → −∂/∂t. There is no change—the equation is invariant under the transformation. The fact that most systems are dissipative is obvious if we run a movie backwards (ink drop, car crash, cigarette smoke...) Dissipation therefore must arise in terms proportional to odd time derivatives; i.e., terms that break time-reversal invariance. In the linear approximation, the damped pendulum equation is d2 θ dθ + γ + ω2θ = 0 2 dt dt where ω 2 = g/l γ = damping coefficient The sign of γ is chosen so that positive damping opposes motion. 5 How does the energy evolve over time? As before, we calculate 1 kinetic energy = ml2 θ̇2 2  2 θ potential energy = mlg(1 − cos θ) ' mlg 2 where we have assumed θ  1 in the approximation. Summing the kinetic and potential energies, we have 1 2  2 g 2 E(θ, θ̇) = ml θ̇ + θ 2 l = 1 2 2 ml (θ̇ + ω 2 θ2 ) 2 Taking the time derivative, 1 dE = ml2 (2θ̇θ̈ + 2ω 2 θ̇θ) dt 2 Substituting the damped pendulum equation for θ̈, dE = ml2 [θ̇(−γ θ̇ − ω 2 θ) + ω 2 θ̇θ] dt = −ml2 γ θ̇2 Take ml2 = 1. Then dE = −γ θ̇2 dt Conclusion: γ = 0 ⇒ Energy conserved (no friction) γ > 0 ⇒ friction (energy is dissipated) γ < 0 ⇒ energy increases without bound 2.2 Phase portrait of damped pendulum Express the damped pendulum as ẋ = θ̇ = y ẏ = θ̈ = −γ θ̇ − ω 2 sin θ = −γy − ω 2 sin x. 6 In the linear approximation, we have      ẋ 0 1 x = . 2 ẏ −ω −γ y The eigenvalues of the system are solutions of (−λ)(−γ − λ) + ω 2 = 0 Thus γ 1p 2 λ=− ± γ − 4ω 2 2 2 2 2 Assume γ  ω (i.e., weak damping, small enough to allow oscillations). Then the square root is complex, and we may approximate λ as γ λ = − ± iω 2 The solutions are therefore exponentially damped oscillations of frequency ω: θ(t) = θ0 e−γt/2 cos(ωt + φ) θ0 and φ derive from the initial conditions. There are three generic cases: • for γ > 0, trajectories spiral inwards and are stable. θ θ • for γ = 0, trajectories are marginally stable periodic oscillations. θ θ 7 • for γ > 0, trajectories spiral outwards and are unstable. θ θ The physical case is the stable case. In the θ, θ̇-phase plane, the phase portrait looks like this: © Informa UK Limited. All rights reserved. Strogatz [3], Fig. 6.7.7 It is obvious from the phase portraits that the damped pendulum contracts areas in phase space: θ θ We can quantify the contraction of areas using the Lie derivative, 1 dV ~ · f~, =∇ V dt which yields ∂ ẋ ∂ ẏ + = 0 − γ = −γ < 0 ∂x ∂y 8 The inequality not only establishes area contraction, but γ gives the rate. 2.3 Summary Finally, we summarize the characteristics of dissipative systems: • Energy not conserved. • Irreversible. • Contraction of areas (volumes) in phase space. Note that the contraction of areas is not necessarily simple. In a 2-D phase space one might expect θ θ time However, we can also have time i.e., we can have expansion in one dimension and (a greater) contraction in the other. In 3-D the stretching and thinning can be even stranger! 9 References 1. Tolman, R. C. The Principles of Statistical Mechanics (Dover, New York, 1979). 2. Bergé, P., Pomeau, Y. & Vidal, C. Order within Chaos: Towards a Deterministic Approach to Turbulence (John Wiley and Sons, New York, 1984). 3. Strogatz, S. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering isbn: 9780429972195 (CRC Press, 2018). 10