1 Toda flows, gradient flows and the generalized Flaschka map Anthony Bloch • Dissipation and Radiation Induced Instability • Toda and gradient flows, metric and metriplectic flows • The Pukhanzky condition • Generalized Flaschka Map • Dispersionless Toda
2 Toda Flow: ˙ X = [ X, Π S X ] Double Bracket Flow: ˙ X = [ X, [ X, N ]] – gradient but special case yields Toda. (with Brockett and Ratiu) ˙ P = [ P, [ P, Λ] (Bloch, Bloch, Flashcka and Ratiu, Total Least Squares).
3 Heat equation u t = u xx Kahler flow: u t = ( − ∆) 1 / 2 u (with Morrison and Ratiu) Dispersionless Toda flow x = { x, { x, z }} ˙ (Bloch, Flaschka, Ratiu) ˙ X = [ X, ∇ H ] + [ X, [ X, N ]] – double bracket dissipation (Bloch, Krishnaprasad, Marsden and Ratiu)
4 • Radiation Damping See Hagerty, Bloch and Weinstein [1999], [2002]. Important early work: Lamb [1900]. Related recent work may be found in Soffer and Weinstein [1998a,b] [1999] and Kirr and Weinstein [2001]. • Original Lamb model an oscillator is physically coupled to a string. The vibrations of the oscillator transmit waves into the string and are carried off to infinity. Hence the oscillator loses energy and is effectively damped by the string. • Lamb model w ( x, t ) displacement of the string. with mass density ρ , ten- sion T . Assuming a singular mass density at x = 0 , we couple dynamics of an oscillator, q , of mass M :
5 Figure 0.1: Lamb model of an oscillator coupled to a string. ∂ 2 w ∂t 2 = c 2 ∂ 2 w ∂x 2 M ¨ q + V q = T [ w x ] x =0 q ( t ) = w (0 , t ) .
6 [ w x ] x =0 = w x (0+ , t ) − w x (0 − , t ) is the jump discontinuity of the slope of the string. Note that this is a Hamiltonian system. Can solve for w and reduce: • Obtain a reduced form of the dynamics describing the explicit motion of the oscillator subsystem, q + 2 T M ¨ c ˙ q + V q = 0 . The coupling term arises explicitly as a Rayleigh dissipation term 2 T c ˙ q in the dynamics of the oscillator.
7 Gyroscopic systems: See Bloch, Krishnaprasad, Marsden and Ratiu [1994]. Linear systems of the form M ¨ q + S ˙ q + Λ q = 0 where q ∈ R n , M is a positive definite symmetric n × n matrix, S is skew, and Λ is symmetric and indefinite. This system Hamiltonian with p = M ˙ q , energy function H ( q, p ) = 1 2 pM − 1 p + 1 2 q Λ q and the bracket { F, K } = ∂F ∂K − ∂K ∂F ∂F ∂K − S ij . ∂q i ∂q i ∂p i ∂p i ∂p i ∂p j Aarise from simple mechanical systems via reduction; normal form of the linearized equations when one has an abelian group.
8 Dissipation induced instabilities—abelian case: Under the above conditions, if we modify the equation to M ¨ q + ( S + ǫR ) ˙ q + Λ q = 0 for small ǫ > 0 , where R is symmetric and positive definite, then the perturbed linearized equations z = L ǫ z, ˙ where z = ( q, p ) are spectrally unstable, i.e., at least one pair of eigenvalues of L ǫ is in the right half plane.
9 • Gyroscopic systens connected to wave fields. ω Figure 0.2: Rotating plate with springs. In Hagerty, Bloch and Weinstein [2002] we describe a gyro- scopic version of the Lamb model coupled to a standard non-
10 dispersive wave equation and to a dispersive wave equation. Show that instabilities will arise in certain mechanical systems. In the dispersionless case, the system is of the form ∂ 2 w ∂t 2 ( z, t ) = c 2 ∂ 2 w ∂z 2 ( z, t ) , � ∂ w � M ¨ q ( t ) + S ˙ q ( t ) + V q ( t ) = T ∂z z =0 w (0 , t ) = q ( t ) , � T is the displacement of the string in the � w = w 1 ( z, t ) · · · w n ( z, t ) first n dimensions and [ ∂ w ∂z ] z =0 is the jump discontinuity in the slope of the string. • Can reduce dynamics to essentially:
11 q ( t ) − V q ( t ) − 2 T M ¨ q ( t ) = − S ˙ c ˙ q ( t ) ,
12 + + + + + + + + + + + + + B Figure 0.3: Gyroscopic Lamb coupling to a spherical pendulum.
13 Metriplectic Systems. A metriplectic system consists of a smooth manifold P , two smooth vector bundle maps π, κ : T ∗ P → TP covering the identity, and two functions H, S ∈ C ∞ ( P ) , the Hamiltonian or total energy and the entropy of the system, such that { F, G } := � d F, π ( d G ) � is a Poisson bracket; in particular π ∗ = (i) − π ; (ii) ( F, G ) := � d F, κ ( d G ) � is a positive semidefinite symmetric bracket, i.e., ( , ) is R -bilinear and symmetric, so κ ∗ = κ , and ( F, F ) ≥ 0 for every F ∈ C ∞ ( P ) ; { S, F } = 0 and ( H, F ) = 0 for all F ∈ C ∞ ( P ) ⇐ (iii) ⇒ π ( d S ) = κ ( d H ) = 0 .
14 Metriplectic Dynamics: d dtF = { F, H + S } + ( F, H + S ) = { F, H } + ( F, S ) Application to Loop Groups...
15 An important and beautiful mechanical system that describes the interaction of particles on the line (i.e., in one dimension) is the Toda lattice. We shall describe the nonperiodic finite Toda lattice following the treatment of Moser. This is a key example in integrable systems theory. The model consists of n particles moving freely on the x -axis and interacting under an exponential potential. Denoting the position of the k th particle by x k , the Hamiltonian is given by n n − 1 H ( x, y ) = 1 � � e ( x k − x k +1) . y 2 k + 2 k =1 k =1
16 The associated Hamiltonian equations are x k = ∂H ˙ = y k , (0.1) ∂y k y k = − ∂H = e x k − 1 − x k − e x k − x k +1 , ˙ (0.2) ∂x k where we use the convention e x 0 − x 1 = e x n − x n +1 = 0 , which corre- sponds to formally setting x 0 = −∞ and x n +1 = + ∞ . This system of equations has an extraordinarily rich structure. Part of this is revealed by Flaschka’s (Flaschka 1974) change of variables given by a k = 1 b k = − 1 2 e ( x k − x k +1 ) / 2 and 2 y k . (0.3)
17 In these new variables, the equations of motion then become a k = a k ( b k +1 − b k ) , ˙ k = 1 , . . . , n − 1 , (0.4) ˙ b k = 2( a 2 k − a 2 k − 1 ) , k = 1 , . . . , n , (0.5) with the boundary conditions a 0 = a n = 0 . This system may be written in the following Lax pair representation: d dtL = [ B, L ] = BL − LB, (0.6) where b 1 a 1 0 ··· 0 0 a 1 0 ··· 0 a 1 b 2 a 2 ··· 0 − a 1 0 a 2 ··· 0 ... , ... . L = B = 0 a n − 1 b n − 1 a n − 1 0 − a n − 1 0 0 a n − 1 b n Can show system is integrable. Generalizations: Lie algebras, rigid body on the Toda orbit (with Gay-Balmaz and Ratiu)
18 More structure in this example. For instance, if N is the matrix diag[1 , 2 , . . . , n ] , the Toda flow (0.6) may be written in the following double bracket form: ˙ L = [ L, [ L, N ]] . (0.7) See Bloch [1990], Bloch, Brockett and Ratiu [1990], and Bloch, Flaschka and Ratiu [1990]. This double bracket equation re- stricted to a level set of the integrals is in fact the gradient flow of the function Tr LN with respect to the so-called normal metric. From this observation it is easy to show that the flow tends asymptotically to a diagonal matrix with the eigenvalues of L (0) on the diagonal and ordered according to magnitude, recovering the observation of Moser, Symes.
19 • Four-Dimensional Toda. Here we simulate the Toda lattice in four dimensions. The Hamiltonian is H ( a, b ) = a 2 1 + a 2 2 + b 2 1 + b 2 2 + b 1 b 2 . (0.8) and one has the equations of motion ˙ b 1 = 2 a 2 a 1 = − a 1 ( b 1 − b 2 ) ˙ 1 , (0.9) ˙ b 2 = − 2( a 2 1 − a 2 a 2 = − a 2 ( b 1 + 2 b 2 ) ˙ 2 ) . (setting b 1 + b 2 + b 3 = 0 , for convenience, which we may do since the trace is preserved along the flow). In particular, Trace LN is, in this case, equal to b 2 and can be checked to decrease along the flow. Figure 0.4 exhibits the asymptotic behavior of the Toda flow.
20 Example 1, initial data [1,2,3,4] 6 4 2 0 a,b −2 −4 −6 −8 0 2 4 6 8 10 12 14 16 18 20 t Figure 0.4: Asymptotic behavior of the solutions of the four-dimensional Toda lattice.
21 It is also of interest to note that the Toda flow may be writ- ten as a different double bracket flow on the space of rank one projection matrices. The idea is to represent the flow in the vari- ables λ = ( λ 1 , λ 2 , . . . , λ n ) and r = ( r 1 , r 2 , . . . , r n ) where the λ i are the i r 2 (conserved) eigenvalues of L and r i , � i = 1 are the top com- ponents of the normalized eiqenvectors of L (see Moser). Then one can show (Bloch (1990)) that the flow may be written as ˙ P = [ P, [ P, Λ]] (0.10) where P = rr T and Λ = diag ( λ ) . This flow is a flow on a simplex The Toda flow in its original variables can also be mapped to a flow convex polytope (see Bloch, Brockett and Ratiu, Bloch, Flaschka and Ratiu).
22 (1,2,3) Schur Horn Polytope Figure 0.5: Image of Toda Flow
23 Toda rigid body: � � (3 D 2 1 − 2 λ ) c 2 1 + 2 dc 1 c 2 + (3 D 1 2 − 2 d ) c 2 h ( c 1 , c 2 , a 1 , a 2 ) = 3 2 � 2 B 1 − B 2 2 B 2 − B 1 � a 2 + a 2 + 6 1 2 2 A 1 − A 2 2 A 2 − A 1 and the associated equations of motion are 2 B 1 − B 2 c 1 = − 2 a 2 (3 D 2 � � ˙ , a 1 = a 1 ˙ 1 − 2 d ) c 1 + dc 2 , 1 2 A 1 − A 2 2 B 2 − B 1 c 2 = − 2 a 2 (3 D 1 � � ˙ , a 2 = a 2 ˙ 2 − 2 d ) c 2 + dc 1 . 2 2 A 2 − A 1
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