LECTURE II. Hamilton-Jacobi theory and Stäckel systems. Maciej B laszak. Poznań University, Poland

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1 LECTURE II Hamlton-Jacob theory and Stäckel systems Macej B laszak Poznań Unversty, Poland Macej B laszak (Poznań Unversty, Poland) LECTURE II 1 / 17

2 Separablty by Hamlton and Jacob Consder Louvlle ntegrable systems on symplectc manfold, gven by {h, h j } π = 0,, j = 1,..., n and related Hamltonan systems u t = X h = πdh, = 1,..., n, u = (q, p) T. (2.1) Assume that (q, p) are canoncal (Darboux) coordnates. HJ method of solvng (2.1) amounts to the lnearzaton of (2.1) va a canoncal transformaton (q, p) (b, a), a = h, = 1,..., n. (2.2) In order to fnd b t s necessary to construct a generatng functon W (q, a) of transformaton (2.2) b = W a, p = W q. The functon W (q, a) s an ntegral of assocated HJ equatons h (q 1,..., q n, W q 1,..., W q n ) = a, = 1,..., n. Macej B laszak (Poznań Unversty, Poland) LECTURE II 2 / 17

3 Separablty by Hamlton and Jacob In (b, a) representaton dynamcal systems (2.1) are trval and (a j ) t = 0, (b j ) t = δ j,, j = 1,..., n, b j (q, a) = W q j = t j + c j, j = 1,..., n. (2.3) Eqs.(2.3) provde mplct solutons of (2.1) n orygnal coordnates. Solvng t for q we reconstruct n explct form trajectores Inverse Jacob problem. q = q (t 1,..., t n, a 1,..., a n, c 1,..., c n ). Where s the hook? How to overcome t? Macej B laszak (Poznań Unversty, Poland) LECTURE II 3 / 17

4 Separablty by Hamlton and Jacob Fnd a dstngushed Darboux coordnates (λ, µ), for whch there exst n relatons (separaton relatons) ϕ (λ, µ, a 1,..., a n ) = 0, = 1,..., n, (2.4) a R, ϕ a j = 0, whch can be solved wth respect to a : a = h (λ, µ), reconstructng our Hamltonans n new coordnates. In fact one can prove that any set of algebrac equatons (2.4) defnes a Lagrangan folaton of symplectc manfold. Macej B laszak (Poznań Unversty, Poland) LECTURE II 4 / 17

5 Separablty by Hamlton and Jacob We are lookng for a generatng functon W (λ, a) of transformaton (λ, µ) (b, a) n the form W (λ, a) = n W (λ, a 1,..., a n ), =1 where functons W are solutons of a system of n decoupled ODEs obtaned from separaton relatons under substtuton µ = dw dλ ϕ (λ, µ = dw dλ, a 1,..., a n ) = 0, = 1,..., n. Such an addtvely separable soluton W (λ, a) s smultanously the soluton of all Hamlton Jacob equatons. (λ, µ) coordnates are called separaton coordnates. Macej B laszak (Poznań Unversty, Poland) LECTURE II 5 / 17

6 Stäckel systems and ther classyfcaton Consder separaton relatons affne n Hamltonans h : n S k (λ, µ )h k = ψ (λ, µ ), = 1,..., n, (2.5) k=1 called generalzed Stäckel separaton relatons. S = ( S k ) - generalzed Stäckel matrx, ψ = (ψ ) - generalzed Stäckel vector. If S k (λ, µ ) = S k (λ, µ ) and ψ (λ, µ ) = ψ(λ, µ ), then (2.5) can be represented by n copes of the curve n S k (λ, µ)h k = ψ(λ, µ) (2.6) k=1 n (λ, µ) plane, called separaton curve. In partcular, f (2.6) s nonsngular, compact Remann Surface Γ, then one can fnd genus of ths curve, basc holomorfc dfferentals and solve the nverse Jacob problem n the language of Remann theta functons. Macej B laszak (Poznań Unversty, Poland) LECTURE II 6 / 17

7 Stäckel systems and ther classyfcaton Let us wrte separaton relatons n the form m [ ] ϕ k (λ, µ ) γ (k) (λ ) + h (k) (λ, n k ) = χ (λ, µ ), = 1,..., n, k=1 (2.7) h (k) n k (λ, n k ) = h (k) j λ n k j, n n m = n, ϕ m (λ, µ ) = 1. j=1 They splt onto bare part m [ ] ϕ k (λ, µ ) E (k) (λ, n k ) = χ (λ, µ ), = 1,..., n, k=1 and generalzed separable potentals m [ ] ϕ k (λ, µ ) γ (k) (λ )δs k + V (k,s) (λ, n k ) = 0, = 1,..., n. k=1 Macej B laszak (Poznań Unversty, Poland) LECTURE II 7 / 17

8 Stäckel systems and ther classyfcaton Basc potentals: γ (s) (λ) = λ r s, r s Z, s = 1,..., m. Then h (k) m + s=1 = E (k) There are m herarches of basc potentals. V (k,s,r s ). Partcular class - fxed Stäckel matrx S and vector χ. S s determned unquely by m vectors ϕ k = (ϕ k 1,..., ϕk n), k = 1,..., m and the partton of n : (n 1,..., n m ). Classcal Stäckel systems: one partcle dynamcs on pseudo-remann space m [ ] ϕ k (λ ) γ (k) (λ ) + h (k) (λ, n k ) = 1 2 f (λ )µ 2, = 1,..., n. k=1 Macej B laszak (Poznań Unversty, Poland) LECTURE II 8 / 17

9 Stäckel systems and ther classyfcaton Benent class: m = 1 γ(λ ) + h 1 λ n h n = 1 }{{} 2 f (λ )µ 2, = 1,..., n h(λ,n) contans majorty of known classcal separable systems n flat or constant curvature Remann spaces, wth all constants of moton quadratc n momenta. Other classes wth m = 1: { γ(λ ) + h 1 λ n h n = }{{} h(λ,n) f (λ )µ 3 exp(aµ ) + exp( bµ ) Perodc Toda, KdV dressng chan, relatvstc n - body problem,... Macej B laszak (Poznań Unversty, Poland) LECTURE II 9 / 17

10 Stäckel systems and ther classyfcaton The case wth m = 2 : ] µ [γ (1) (λ ) + h (1) (λ, n 1 ) [ ] + γ (2) (λ ) + h (2) (λ, n 2 ) = f (λ )µ 3, statonary flows of Bussnesq: n 1 = 1, n 2 = n 1, dynamcal system on loop algebra sl(3) : n 1 = 2, n 2 = 4. Macej B laszak (Poznań Unversty, Poland) LECTURE II 10 / 17

11 Classcal Stäckel systems (Q, g) pseudo Remann space and dynamcal system q tt + Γ jk qj tq k t = G k k V (q), = 1,..., n (2.8) q t = h p, (p ) t = h q, h(q, p) = 1 2,j G j p p j + V (q). Assume that (2.8) s Louvlle ntegrable wth all constants of moton quadratc n momenta: h r (q, p) = 1 2,j (K r G ) j p p j + V r (q), r = 1,..., n, where K 1 = I, h 1 = h, K r Kllng tensors. The transformaton to separaton coordnates (λ, µ) s a pont transformaton generated by Macej B laszak (Poznań Unversty, Poland) LECTURE II 11 / 17

12 Classcal Stäckel systems F (p, λ) = n θ (λ)p = q = F = θ (λ), =1 p µ = F λ = p θ (λ) λ. The explct form of Hamltonans n separaton coordnates depends on the form of separaton relatons. Example. Benent class: G j = f (λ ) δ j, (K r ) j = ρ r λ δ j, V r = ρ r γ (λ ), λ where = (λ λ k ) k = and ρ r are Vete polynomals: ρ 1 = (λ λ n ),..., ρ n = ( 1) n λ 1...λ n. Macej B laszak (Poznań Unversty, Poland) LECTURE II 12 / 17

13 Classcal Stäckel systems Lnearzaton n (b, a) coordnates. Consder subclasses λ k + h 1 λ γ 1 + h 2 λ γ h n = 1 2 f (λ )µ 2, = 1,..., n (2.9) and generatng functon W (a, λ) = W (λ, a) µ = dw, dλ Hence, from (2.9) 1 2 f (λ ) b = W a = ( ) 2 dw = λ k dλ n λj j=1 n + r=1 ξ γ dξ Rj (ξ, a) = t + c, b = W a. a r λ γ r P(λ, a) = 1,..., n Macej B laszak (Poznań Unversty, Poland) LECTURE II 13 / 17

14 Classcal Stäckel systems where R j (ξ, a) = 2f j (ξ)p(ξ, a), or n Abel-Jacob dfferental form n j=1 λ γ j dλ j Rj (λ j, a) = dt, = 1,..., n. Examples. Natural Hamltonans of one-degree of freedom h = 1 2m p2 + V (x). (2.10) (x, p) are separaton coordnates and (2.10) tselve represents separaton relaton. Take harmonc oscllator wth V (x) = 1 2 αx 2. Then ( ) 1 dw m dx 2 αx 2 = a = W = 2m(a 1 2 αx 2 )dx Macej B laszak (Poznań Unversty, Poland) LECTURE II 14 / 17

15 Classcal Stäckel systems = t + c = dw da = mdx = 2m(a 1 2 αx 2 ) m α α arcsn 2a x = x = A sn(ωt + ϕ), A = 2a α α, ω = m, ϕ = ωc. For potental V (x) = 1 2 αx βx 4 the smlar calculaton gves x = Asn(ωt + ϕ, k) snus elptc functon of Jacob, where (A, ω, k) are expressed by (α, β, a) through mβa 2 = 2k 2 ω 2, mα = ω 2 (1 + k 2 ), 2a = A 2 ω 2, ϕ = ωc. Macej B laszak (Poznań Unversty, Poland) LECTURE II 15 / 17

16 Classcal Stäckel systems Henon-Heles system (two degrees of freedom) h h 1 = 1 2 p p2 2 + q q 1q 2 2, h 2 = 1 2 q 2p 1 p q 1p q2 1q q2 2. Transformaton to separaton coordnates q 1 = λ 1 + λ 2, p 1 = λ 1µ 1 + λ 2µ 2, λ 1 λ 2 λ 2 λ 1 q 2 = 4λ 1 λ 2, p 1 = ( µ1 λ 1 λ 2 + µ ) 2. λ 1 λ 2 λ 2 λ 1 Macej B laszak (Poznań Unversty, Poland) LECTURE II 16 / 17

17 Classcal Stäckel systems Separaton relatons h 1 λ 1 + h 2 = 1 2 λ 1µ λ 4 1 h 1 λ 2 + h 2 = 1 2 λ 2µ λ 4 2. Implct soluton n Abel-Jacob form dt 1 = λ 1dλ 1 R(λ1, a) + λ 2dλ 2 R(λ2, a) dt 2 = dλ 1 R(λ1, a) + dλ 2 R(λ2, a), where R(λ, a) = a 2 + a 1 λ λ 4. Macej B laszak (Poznań Unversty, Poland) LECTURE II 17 / 17