THE BLOCH-ISERLES SYSTEM Tudor S. Ratiu ematiques and Bernoulli Center Section de Math´ Ecole Polytechnique F´ ed´ erale de Lausanne, Switzerland tudor.ratiu@epfl.ch Joint work with A. Bloch, V. Br ˆ ınz˘ anescu, A. Iserles, J. Marsden F. Gay-Balmaz Bi-Hamiltonian Systems and All That, September 27–October 1, 2011 1
PLAN OF THE PRESENTATION • A new system on symmetric matrices • Lie algebra structure on the space of symmetric matrices • The Mishchenko-Fomenko free rigid bodies • Analysis of the Poisson structures • Relation to the Mischenko-Fomenko systems • Lax pair with parameter • Bi-Hamiltonian structure and integrability • Linearization of the flows Bi-Hamiltonian Systems and All That, September 27–October 1, 2011 2
A NEW SYSTEM ON SYMMETRIC MATRICES Bloch and Iserles [2006] have introduced the system for X ∈ Sym( n ) X = [ X 2 , N ] = [ X, XN + NX ] , ˙ N ∈ so ( n ) constant . Since [ X 2 , N ] ∈ Sym( n ), so X (0) ∈ Sym( n ) ⇒ X ( t ) ∈ Sym( n ), ∀ t . � � X, Y � � := trace ( XY ), X, Y ∈ Sym( n ) is positive definite. Function 1 2 trace( X 2 ) = 1 2 � � X, X � � X = [ X 2 , N ] has solutions ∀ t ∈ R . ˙ conserved, compact level sets so Reasons given for studying this system: ˙ 1.) X = [ X, N ] X + X [ X, N ] a special case of a congruent flow X = A ( X ) X + XA ( X ) T , ˙ A : Sym( n ) → gl ( n, R ) smooth Solution is X ( t ) = V ( t ) X (0) V ( t ) T , where V ( t ) is the solution of � V ( t ) X (0) V ( t ) T � ˙ V ( t ) = A V ( t ) , V (0) = I. Bi-Hamiltonian Systems and All That, September 27–October 1, 2011 3
So the solution is given by the action of GL( n, R ) on Sym( n ) by congruence. In particular, the signature of X ( t ) is conserved. ˙ 2.) X = [ X, XN + NX ], B ( X ) := XN + NX , is isospectral. The solution is given by the SO( n )-action on Sym( n ) by similarity: X ( t ) = Q ( t ) − 1 X (0) Q ( t ), where Q ( t ) ∈ SO( n ) is the solution of � Q ( t ) − 1 X (0) Q ( t ) � ˙ Q ( t ) = Q ( t ) B , Q (0) = I. X = [ X 2 , N ] = [ X, NX + XN ] is evolving under two ˙ So the system distinct group actions. M = [ M, Ω] = [Ω 2 , J ], where ˙ 3.) “Dual” to the SO( n ) rigid body M = J Ω + Ω J for J = diag( J 1 , . . . , J n ) with J i + J j > 0 for i � = j . 4.) Numerically: regular behavior as for an integrable system. EXAMPLE 1. 3 × 3 case with N 12 = N 23 = 1, so N invertible with ( x 12 , x kl ) for ( k, l ) = (1 , 1) , (1 , 3) , (2 , 2) , (2 , 3) , (3 , 3) with a random initial condition; Bloch-Iserles [2005]. Bi-Hamiltonian Systems and All That, September 27–October 1, 2011 4
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EXAMPLE 2. Lowest dimension degenerate case 2 p + d = 3 with p = 1 , d = 1. Let a e f � � S A X = e b g =: A T c f g c and � ¯ 0 1 0 � N 0 N = − 1 0 0 =: . 0 0 0 0 0 X = [ X 2 , N ] is ˙ Then the dynamics ˙ a = − 2( ae + eb + fg ) , ˙ b = 2( ae + eb + fg ) , c = 0 , ˙ e = a 2 + f 2 − b 2 − g 2 , ˙ ˙ g = af + ge + cf, ˙ f = − ( ef + bg + gc ) . We shall see later that the Bloch-Iserles system is Lie-Poisson. The two Casimir functions are C 2 = c , so that ˙ c = 0 and − ba + g 2 a + f 2 b � � C 1 = 1 + e 2 − 2 fge = − det X 2 2 c c c c Can check directly that C 1 , C 2 are conserved. Bi-Hamiltonian Systems and All That, September 27–October 1, 2011
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 0 10 20 30 40 50 60 70 80 90 100 trace( X ) and trace( X 2 ) are conserved. Conservation of trace( X ) is given by adding the first two equations of motion while trace( X 2 ) / 2 is the Hamiltonian. Bi-Hamiltonian Systems and All That, September 27–October 1, 2011 6
Time plot of flow in the 3 × 3 degenerate case for a , b , c , e , f , g Phase plane portraits projected to the a - e and the b - e planes 0.8 0.7 0.6 0.6 0.5 0.4 0.4 0.2 0.3 0 0.2 −0.4 −0.2 0 0.2 0.4 0 0.2 0.4 0.6 0.8 Bi-Hamiltonian Systems and All That, September 27–October 1, 2011 7
5.) Fundamental link with plasma phyiscs; Holm-Tronci [2009] Poisson-Vlasov system � � f, δH ˙ f + = 0 δf can where { , } can is the canonical Poisson bracket on R 2 n , f ( q , p , t ) is the plasma density, H ( f ) := 1 � � p � 2 + U f ( q , t ) � � f ( q , p , t ) d q d p 2 is the Hamiltonian and U f is the nonlinear collective potential de- � f ( q , p , t )d p . termined by ∆ U f ( q , t ) = Hamiltonian system relative to the Lie-Poisson bracket on F ( R 2 n ) � � δF δf , δG � { F, G } ( f ) = f ( q , p ) d q d p δf can What is the group? Bi-Hamiltonian Systems and All That, September 27–October 1, 2011 8
Naive: all symplectic diffeomorphisms of R 2 n . But this implies that we identify Hamiltonian vector fields with their Hamiltonians, which is incorrect. The correct group is formed by contactomorphisms Cont( R 2 n +1 , θ ) := ϕ ∈ Diff( R 2 n +1 ) | ϕ ∗ θ = θ � � , θ := p · dq − d s whose Lie algebra is ∼ cont ( R 2 n +1 , θ ) := � X ∈ X ( R 2 n +1 ) | £ X θ = 0 � → θ ( X ) ∈ F ( R 2 n ) ∋ X �− θ ( X ) is independent of s ; van Hove [1951] (thesis) isomorphism Geodesic Vlasov equations: ˙ f + { f, G ⋆ f } can = 0 G ( z , z ′ ) a kernel z := ( q , p ) and ⋆ is convolution. H ( f ) = 1 f ( z ) G ( z , z ′ ) f ( z ′ )d z d z ′ = 1 �� 2 � f � 2 G 2 Bi-Hamiltonian Systems and All That, September 27–October 1, 2011 9
m th statistical moment : contravariant symmetric tensor on R 2 n m X m ( f ) := 1 � R 2 n , z ⊗ m f ( z )d z ∈ � m = 0 , 1 , 2 , . . . i.e., m ! i =0 X m ( f ) i 1 ··· i m := 1 � z i 1 · · · z i m f ( z ) d z , z = ( z 1 , . . . , z n ) m ! f �→ { X m ( f ) } to the moments takes the Vlasov Lie-Poisson bracket �� m i =0 R 2 n � to the Lie-Poisson bracket on moments ST ( R 2 n ) := ⊕ ∞ m =0 � ∂F ∞ , ∂G � �� � { F, G } ( X ) = X n + m − 2 , ∂X n ∂X m n,m =0 where � , � is given by tensor contraction, the moment Lie bracket on symmetric contravariant tensors is ( n + m − 2)! � sym [ S n , S ′ � S n J − 1 S ′ m ] := − m ( n − 1)!( m − 1)! A sym is the symmetric part of the tensor A , ( AB ) hk... ij... := A ij...p B phk... Bi-Hamiltonian Systems and All That, September 27–October 1, 2011 10
� � 0 1 = − J − 1 = − J T J = − 1 0 m =0 S m z ⊗ m ∈ Holm, Lysenko, Scovel [1990]: ST ( R 2 n ) ∋ { S m } �→ � ∞ F ( R 2 n ) is a Lie algebra homomorphism and [ ST m ( R 2 n ) , ST p ( R 2 n )] ⊂ ST m + p − 2 ( R 2 n ). So ST 2 ( R 2 n ) = Sym( R 2 n ) = R 2 n ∨ R 2 n subalgebra. Consider three moments in Sym( R 2 n ) ⊕ R 2 n ⊕ R . Lie-Poisson bracket � � ∂F , ∂G { F, G } ( X 2 , X 1 , X 0 ) = Tr X 2 ∂X 2 ∂X 2 J � � ∂F J ∂G − ∂G J ∂F + X T 1 ∂X 1 ∂X 2 ∂X 1 ∂X 2 � T � ∂F ∂G + X 0 J ∂X 1 ∂X 2 where � � � z ⊗ 2 f ( z )d z X 0 = f ( z )d z , X 1 = z f ( z )d z , X 2 = Bi-Hamiltonian Systems and All That, September 27–October 1, 2011 11
The function h ( X 2 , X 1 , X 0 ) = 1 2 ) + 1 4 � X 1 � 2 + 1 2 Tr( X 2 8 X 2 0 comes from the quadratic Vlasov Hamiltonian H ( f ) = 1 G ( z , z ′ ) = 1 8+1 4 z · z ′ +1 �� f ( z ) G ( z , z ′ ) f ( z ′ )d z d z ′ 8( z · z ′ ) 2 for 2 We shall see: h is a Bloch-Iserles Hamiltonian for N of corank 1. Gay-Balmaz & Tronci [2011] have determined the underlying group. Will be discussed later. LIE ALGEBRA STRUCTURE ON SYMMETRIC MATRICES (Sym( n ) , [ , ] N ) is a Lie algebra: [ X, Y ] N := XNY − Y NX ∈ Sym( n ) N can be thought of as a Poisson tensor on R n : { f, g } N = ( ∇ f ) T N ∇ g or X h = N ∇ h Bi-Hamiltonian Systems and All That, September 27–October 1, 2011 12
Each X ∈ Sym( n ) defines the quadratic Hamiltonian Q X by Q X ( z ) := 1 z ∈ R n . 2 z T Xz, Define Q := { Q X | X ∈ Sym( n ) } . Then Q : X ∈ Sym( n ) �→ Q X ∈ Q is a linear isomorphism. The Hamiltonian vector field of Q X has the form X Q X = NX and the Poisson bracket of two such quadratic functions is { Q X , Q Y } N = Q [ X,Y ] N , ∀ X, Y ∈ Sym( n ) Q : X ∈ (Sym( n ) , [ , ] N ) �→ Q X ∈ ( Q , { , } N ) Lie algebra isomorphism. Bi-Hamiltonian Systems and All That, September 27–October 1, 2011 13
LH = the Lie algebra of linear Hamiltonian vector fields on R n relative to the commutator bracket of matrices X ∈ (Sym( n ) , [ · , · ] N ) �→ NX ∈ ( LH , [ · , · ]) is a Lie algebra homomorphism and if N is invertible (so n is even) it induces an isomorphism of (Sym( n ) , [ · , · ] N ) with sp ( n, N − 1 ) := { Z ∈ gl ( n ) | Z T N − 1 + N − 1 Z = 0 } ( u , v ) �→ u · N − 1 v symplectic form. The key identity in the proof is: N [ X, Y ] N = [ NX, NY ] , X, Y ∈ Sym( n ) Bi-Hamiltonian Systems and All That, September 27–October 1, 2011 14
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