Quantum integrable systems and Geometry September 3-7, 2012, Olhao Analytic prolongation of normal forms JP Fran¸ coise Universit´ e P.-M. Curie, Paris 6, France 1
Birkhoff normal forms G. D. Birkhoff studied the local expression of a Hamiltonian system near a critical point of Morse type up to symplectic changes of coordinates. Under some generic conditions, this local normal form exists as a formal series in any dimension. It is convergent in one degree of freedom and it is generically divergent in ( m > 1) degrees of freedom [(Siegel, 54)(Moser, 76)] 2
Hamiltonian systems integrable in Liouville sense A Hamiltonian system ( H, ω ) is said to be (completely) integrable if there exist m generically independent integrals H = ( H 1 , ..., H m ) such that { H i , H j } = 0 , H = H 1 . If the fibers H − 1 ( c ) are compact and connected, they are torii and the flows of all the H i are linear on these torii. Action- angles coordinates allow to compute the frequencies of the Hamiltonian flow of H on these invariant torii. 3
Analytic Hamiltonian systems which are Liouville integrable display a convergent Birkhoff normal form Theorem (J. Vey, 76) Assume H are analytic near 0 ∈ R n , { H i , H j } = 0 , H = H 1 displays a Morse critical point, assume that the Hess H i (0) generate a Cartan sub- algebra of Sp (2 m, R ) , then the Birkhoff normal form of H is a convergent series. For instance, p i = x 2 i + y 2 i or p i = x i y i generate a Cartan sub-algebra of Sp (2 m, R ) (Precise definition : commutative and selfnormalizing). 4
Birkhoff normal form of an integrable Hamiltonian system is in general a convergent series but a priori only defined in the neighborhood of the critical point. What can be said of its analytic prolongation if the Hamiltonian system is itself globally defined (for instance is a polynomial, rational function) ? In such case, if the HS displays different critical points, is it possible to compare the analytic prolongation of the Birkhoff form in one critical point to the Birkhoff form in the other critical point ? Begin with one degree of freedom ! 5
The pendulum has been studied recently (P.L.Garrido, G. Gallavotti, JPF, Journal Maths Physics 10). Reading Jacobi in the text and Gradshtein-Ryzhik tables of formula for elliptic functions. Jacobi found a coordinate system in which the motion is linear but he did not computed the symplectic form in these coordinates. Although his computation could be used to obtain a piece of information on the symplectic form (its relative cohomology class associated with H ). 6
Singularity theory of functions 2 ( x 2 + y 2 ) + .... and a Consider an analytic function H : ( x, y ) �→ 1 symplectic (volume) form ω = dx ∧ dy . Morse lemma allows to find anew 2 ( X 2 + Y 2 ) but with (analytic) coordinate system ( X, Y ) such that H = 1 no control of ω : ω = [1 + F ( X, Y )] dX ∧ dY . Definition Two volume forms ω and ω ′ have the same relative cohomo- logy class if ω − ω ′ = dH ∧ dξ . Moser isotopy method for volume form applies and shows : Given a function H and two volume forms which are relatively cohomologous, there is an isotopy φ so that φ ∗ ( H ) = H and φ ∗ ( ω ) = ω ′ . Any polynomial form ω decomposes into ω = ψ ( H ) dx ∧ dy + dH ∧ dξ . 7
2 ( x 2 + y 2 ) , ω = 1 After these changes of coordinates we find H = ψ ( H ) dx ∧ dy , there is and easy change into, H = φ (1 2( x 2 + y 2 )) , ω = dx ∧ dy and this is the Birkhoff normal form ! 8
Free rigid body motion and the geodesic motion on a revolution ellipsoid have been also studied (P.L. Garrido, G. Gallavotti, JPF, F18 in Ipparco Roma I, 2012.) Normal forms are derived via the analysis of relative cohomology. 9
The Hamiltonian of the free rigid body in the coordinates B, β depending of the parameters I and of the ”parameter” A (in fact constant of motion) : + sin 2 β B 2 2(cos 2 β H ′ = 1 + 1 )(( A 2 − B 2 ) , (1) 2 I 3 I 1 I 2 We consider instead : H = 2 H ′ − A 2 I − 1 = B 2 − r 2 sin 2 βA 2 + r 2 sin 2 βB 2 , 1 (2) I − 1 − I − 1 3 1 with r 2 = I − 1 − I − 1 1 2 . (3) I − 1 − I − 1 3 2 10
Consider the Hamiltonian : H = B 2 + r 2 β 2 A 2 + r 2 (sin 2 β − β 2 ) A 2 − r 2 sin 2 βB 2 . (4) Change ( B, β ) into ( X = B, Y = rAβ ) , this multiplies the symplectic form by 1 /rA and this yields to consider in the following the couple : H = X 2 + Y 2 + ...., ω = dX ∧ dY (5) 11
We have here an explicit version of the Morse lemma : ( X, Y ) �→ ( X ′ , Y ′ ) � X ′ = X 1 − r 2 sin 2 ( Y (6) rA ) = X + ...h.o.t. Y ′ = rA sin( Y rA ) = Y + ...h.o.t. In these coordinates : H = X ′ 2 + Y ′ 2 , 1 dX ′ ∧ dY ′ . ω = dX ∧ dY = (7) � (1 − Y ′ 2 A 2 )(1 − Y ′ 2 r 2 A 2 ) 12
We compute the cohomology class of the volume form ω by introducing the coefficients a k defined as follows : 1 1 − u 2 = Σ k a k u 2 k , √ (8) and the binomial coefficients C k n . One can check that ω = g ( ξ ) dX ′ ∧ dY ′ + dξ ∧ du, (9) where g ( ξ ) = Σ h (Σ k,l ; k + l = h a k a l r 2 h ) C h ξ h 2 h r 2 h . (10) 4 h 13
It remains to perform a final change of coordinates of type : x = X ” u ( ξ ) , y = Y ” u ( ξ ) , so that U ( ξ ) = u 2 ( ξ ) satisfies : U ( ξ ) + ξU ′ ( ξ ) = g ( ξ ) . Note that this last equation is easily solved in terms of formal series, if g n g ( ξ ) = Σ n g n ξ n , then U ( ξ ) = Σ n n +1 .Inverting the series : ξ = HU ( H ) : C h H h U ( H ) = Σ h (Σ k + l = h a k a l r 2 l ) 2 h r 2 h 4 h ( h + 1) into H = ξV ( ξ ) yields the Birkhoff normal form. 14
The coefficients of the Birkhoff normal form are polynomials in a variable r depending on the inertia moments. We checked numerically that their roots are on the unit circle. We proved that this is true for the series issued from the chomology class and discuss the link with D. Ruelle’s articles on the extensions of the Lee-Yang theorem. 15
Computation of canonical partition functions This second part closely relates to a question posed by G. Gallavotti and C. Marchioro in 1973. Let us denote by ∆ a root system of rank r. It is a set of vectors of R r which is invariant under reflections in the hyperplane perpendicular to each vector in ∆ . A reflection s ρ in terms of a root ρ is defined by s ρ ( x ) = x − ρ (2 ρ.x ) /ρ 2 . Thus ∆ is characterized by s ρ ( η ) ∈ ∆ , ρ, η ∈ ∆ 16
The dynamical variables are the coordinates q i , i = 1 , ..., r. and their canonically conjugate momenta p i , i = 1 , ..., r. The Hamiltonian for the generalized Calogero system with an external quadratic potential (Bordner- Corrigan-Sasaki, 00) is : g | ρ | | ρ | 2 H = 1 2 p 2 + 1 + 1 2 ω 2 q 2 2Σ ρ ∈ ∆+ ( ρ.q ) 2 in which the coupling constants g | ρ | are defined on orbits of the correspon- ding Coxeter group. 17
Choose a representation E of dimension D of the Coxeter group, then define the D × D matrix : p. ˆ H : ( p ˆ H ) αβ = ( p.β ) δ αβ where α and β are vectors belonging to the representation. Introduce next the D × D matrices X, L and M : H )1 X = iΣ ρ ∈ ∆+ g | ρ | ( ρ ˆ ρ.q ˆ s ρ L = p ˆ H + X ( ρ ) 2 M = i 2Σ ρ ∈ ∆ + g | ρ | ( ρ.q ) 2 ˆ s ρ 18
where s ραβ = δ α,s ρ ( β ) , ˆ and a diagonal matrix : Q = q ˆ H, Q αβ = ( q.α ) δ αβ The time evolution of the Hamiltonian displays ˙ L = [ L, M ] − ω 2 Q ˙ Q = [ Q, M ] + L Introduce next : L ± = L ± i ωQ with P = L + L − , 19
we get a Lax Matrix : ˙ P = [ P, M ] Consider the symplectic form : Tr ( dQ ∧ dL ) = C D Σ r j =1 dq j ∧ dp j defined on the product of two copies of the representation. The constant C D depends actually on the representation. Let Λ be an eigenvalue of the matrix P and let T be the matrix of the projection onto the eigenspace corresponding to this eigenvalue. Classical result of linear perturbation theory yields : d Λ = Tr ( TdP ) 20
The Hamiltonian flow generated by the function Λ and the symplectic form yields : ˙ Q = [ Q, M ] + i ω [ T, Q ] + ( LT + TL ) , ˙ L = [ L, M ] + i ω [ T, L ] − ω 2 ( QT + TQ ) , and thus : ˙ P = [ P, M ] + 2i ω [ T, P ] = [ P, M ] This shows that the eigenvalues of the Lax matrix P are constants of motion for the Hamiltonian flow generated by any of its eigenvalues. In particular this proves that the Hamiltonian flows generated by the eigenvalues of the Lax matrix P Poisson commute. The Hamiltonian flows generated by any eigenvalues of the Lax matrix P have all orbits periodic of the same period π/ω . 21
Introduce U the (time dependent) matrix solution of the Cauchy pro- blem : ˙ U = UM, U (0) = I The matrix UPU − 1 is then a constant of the motion. Denote V a time- independent matrix which diagonalizes this matrix. Conjugate all the ma- trices UL ± U − 1 , UTU − 1 by the matrix V yields : L ′± = ± 2i ωτL ′± ˙ where L ′± = V UL ± U − 1 V − 1 and τ is the constant diagonal matrix whose entries are equal to zero except the diagonal term equals to 1 in the position corresponding to the eigenvalue. These equations can be easily integrated and they yield the periodicity of the eigenvalues of the matrix 2i ω ( L + − L − ) At this point we have obtained that there is a familly of 1 Q = commuting flows which are all isochronous. (Caseiro, JPF, Sasaki, JMP00) 22
Recommend
More recommend
