Liouville integrability of Hamiltonian systems and spacetime symmetry

Transcription

1 Seminar, Kobe U., April 22, 2015 Liouville integrability of Hamiltonian systems and spacetime symmetry Tsuyoshi Houri with D. Kubiznak (Perimeter Inst.), C. Warnick (Warwick U.) Y. Yasui (OCU Setsunan U.)

2 Hamilton formalism Many dynamical systems in physics are described in this framework. A dynamical system is governed by a function of canonical coordinates q and momenta p, called Hamiltonian H(q, p ) Equations of motion dq dτ = H p dp dτ = H q

3 Hamilton formalism Poisson bracket A, B A q B p A p B q Conserved quantity (First integral) df dτ = F dq q dτ + F dp p dτ = F q H p F p H q = F, H F is a conserved quantity F, H =0

4 Hamilton formalism Liouville integrability If there exist D independent Poisson-commuting constants α i (including Hamiltonian) in a D-dim Hamiltonian system, then the system is said to be completely integrable. Namely, one can prove that there exists a canonical transf. (x, p) (φ, I(α)), and then easily solve the Hamilton s eq.:

5 Hamiltonian H = 1 2, g (q)p p + V(q) g (q) : metric (q, p ) : canonical coordinates ds = g dq dq V(q) : potential V 0 V = 0 Natural Hamiltonian Geodesic Hamiltonian

6 The purpose of this talk To show a systematic approach for investigating polynomial conserved quantities for any natural Hamiltonian system

7 Keywords 1 Geometrisation Any natural Hamiltonian system can be translated to the geodesic problem in a corresponding spacetime. ➁ Prolongation Equations describing spacetime symmetry can be translated to a first-order linear PDE system.

8 Geodesic problem and spacetime symmetry

9 Spacetime symmetry Isometry Killing equation invariant Constant along geodesics geodesic

10 Spacetime symmetry Conserved quantities along geodesics

11 For a geodesic Hamiltonian H = g p p, when F is a n-order homogeneous polynomial in p, then we find that Killing-Stackel Eq.

12 Spacetime symmetry Conserved quantities along geodesics Isometry (Killing vector) Hidden symmetry (Killing tensor) first-order polynomial higher-order polynomial

13 Spacetime symmetry Killing vector fields: ξ μ ξ ν + ν ξ μ = 0 Killing-Stackel tensors (μ K ν1 ν 2 ν n ) = 0 Killing-Yano tensors (μ f ν1 )ν 2 ν n = 0 [Stackel 1895] [Yano 1952] K (μ1 μ 2 μ n ) = K μ1 μ 2 μ n f [μ1 μ 2 μ n] = f μ1 μ 2 μ n

14 vector fields Killing Conformal Killing symmetric anti-symmetric Killing-Stackel Killing-Yano Stackel 1895 Yano 1952 Conformal Killing-Stackel Conformal Killing-Yano Tachibana 1969, Kashiwada 1968

15 Why spacetime symmetry? Conserved quantities along geodesics Integrability of EOMs for matter fields Klein-Gordon and Dirac equations Classification of spacetimes Stationary, axially symmetric, Bianchi type, etc. Application to Hamiltonian dynamics

16 Geometrisation

17 Basic idea The dynamical trajectories of a Hamiltonian system of the form can be seen as geodesics of a corresponding configuration space, or of enlargement of it, under some constraints.

18 Examples 1 Maupertuis principle ➁ Canonical transformations Ex. ➁-1 3D Kepler problem Ex. ➁-2 N=3 open Toda ➂ Eisenhart s lifts Ex. ➂-1 Eisenhart lift Ex. ➂-2 Generalised Eisenhart lift Ex. ➂-3 Light-like Eisenhart lift

19 Maupertuis principle Maupertuis 1744, 1746, 1756 One obtains an integral equation that determines the path followed by a physical system without specifying the time parameterization of that path. Newtonian y x (t) GR t x (τ) x x

20 Maupertuis principle Maupertuis 1744, 1746, 1756 One obtains an integral equation that determines the path followed by a physical system without specifying the time parameterization of that path. q = q(t), p = p(t) action S = p dq H q, p dt abbreviated action S (E) = p dq

21 Jacobi s formulation Lagrangian L = 1 2, g q q U(q) momentum p L q = energy E = 1 2, g q g q q + U(q) abbreviated action S p dq =, g dq dt dq = dt = 2 E U g dq dq, ( ) g = E U g S = 2, g dq dq

22 Jacobi s formulation

23 Jacobi s formulation Theorem Given a dynamical system on a manifold (M, g ) i.e., a dynamical system whose Lagrangian is then it is always possible to find a conformal transformation of the metric (Jacobi metric) such that the geodesics of (M, g ) with the energy E = 1 are equivalent to the trajectories of the original dynamical system.

24 Comparison of Hamiltonians Natural Hamiltonian H = 1 2 g p p + U with H = E Jacobi s Hamiltonian equivalent! E = H + U H = 1 2 g p p with g = E U g H = 1 1 = H E U

25 Comparison of first integrals Natural Hamiltonian Jacobi s Hamiltonian For instance,

26 Maupertuis principle Potential system Geodesic system

27 Natural Hamiltonian Qudartic + potential Geodesic Hamiltonian H = 1 2 g (q)p p + U(q) Homogeneously quadratic H = 1 2 g q p p Point! We need to construct a geodesic Hamiltonian, i.e., a homogeneously quadratic Hamiltonian which reduces to the original natural Hamiltonian under some transformation or constraints.

28 Examples 1 Maupertuis principle ➁ Canonical transformations Ex. ➁-1 3D Kepler problem Ex. ➁-2 N=3 open Toda ➂ Eisenhart s lifts Ex. ➂-1 Eisenhart lift Ex. ➂-2 Generalised Eisenhart lift Ex. ➂-3 Light-like Eisenhart lift

29 Ex D Kepler problem Keane-Barrett-Simmons 2000 H = 1 2 p + p + p α r with H = E r = q + q + q Transf. q = p, p = q H = E 1 2 r ( p + p + p ) with H = α r = q + q + q ds = E 1 2 r d q + d q + d q (constant curvature 4E )

30 Ex. 2-2 N=3 open Toda Baleanu-Karasu-Makhaldiani 1999 H = 1 2 p + p + p + a + a a = e, a = e Transf. q = q + ln p, q = q, q = q ln p p = p, p = p, p = p H = a p + p a p a = e, a = e ds = d q a + d q + d q a

31 Examples 1 Maupertuis principle ➁ Canonical transformations Ex. ➁-1 3D Kepler problem Ex. ➁-2 N=3 open Toda ➂ Eisenhart s lifts Ex. ➂-1 Eisenhart lift Ex. ➂-2 Generalised Eisenhart lift Ex. ➂-3 Light-like Eisenhart lift

32 Ex. 3-1 (Standard) Eisenhart lift Natural Hamiltonian on n-dim space (M, g) H = 1 2 g (q)p p + U(q) Geodesic Hamiltonian on (n+1)-dim space (M x R, g E ) H = 1 2 g p p + U q p Eisenhart metric = 1 2 g p p ds = 2U q ds + g dq dq

33 Ex. 3-2 Generalised Eisenhart lift Natural Hamiltonian on n-dim space (M, g) H = g (q)p p + U(q), U q = ll a ll U ll (q) Geodesic Hamiltonian on (n+m)-dim space (M x R m, g E ) Eisenhart metric H = 1 2 g p p + = 1 2 g p p ll U ll q p ll ds = 2U ll q (ds ll ) + g dq dq

34 Ex. 3-3 Light-like Eisenhart lift Natural Hamiltonian on n-dim space (M, g) H = 1 2 g (q)p p + U(q) Geodesic Hamiltonian on (n+m)-dim spacetime (M x R m, g E ) H = 1 2 g p p + U q p + p p Eisenhart metric = 1 2 g p p ds = 2U q dt + 2dt ds + g dq dq

35 Comparison Natural Htn v.s. LL Eisenhart s Htn Natural Hamiltonian Eisenhart s Hamiltonian H = H + U H = H + Up + p p For instance,

36 Prolongation

37 Review I: Integrability conditions for systems of first order PDEs

38 A system of first order PDEs u x = ψ x u u = (u, u,, u ) ; unknown functions x = (x, x,, x ) ; variables Questions : Does the solution exist? If exist, is the solution space finite or infinite? How many dimensions? Explicit expressions? at most N dimensions

39 Consistency conditions u u x x x x = 0 u x x = ψ u x x = ψ x x u + ψ u x = ψ x u + ψ ψ u + ψ u x = ψ x u u + ψ ψ u ψ x ψ x + ψ ψ ψ ψ u = 0

40 Frobenius theorem The necessary and sufficient conditions for the unique solution u = u (x) to the system u x = ψ u such that u x = u to exist for any initial data (x, u ) is that the relation hold. ψ x ψ x + ψ ψ ψ ψ = 0

41 Parallel equation u π (p) u x = ψ x u E u x ψ x u = 0 π p M D u = 0 where D u : = u x ψ x u The system can be viewed as a parallel equation for sections u of a vector bundle π: E M of rank N.

42 Curvature conditions For a connection D D u : = u x ψ x u the curvature of D is defined by (D D D D )u = R u. D u = 0 R u = 0 This is equivalent to the Frobenius integrability condition

43 Frobenius theorem (II) The necessary and sufficient conditions for the unique solution u = u (x) to the system where such that u x = u to exist for any initial data (x, u ) is that the relation hold. D u = 0 D u : = u x ψ R u = 0 i = 1,, n x u α = 1,, N

44 Discussion If the curvature conditions hold, the general solution depends on N arbitrary constants. u x; a = a u (x) + a u (x) + + a u (x) If not, they give a set of algebraic equations R u = 0 Differentiating these equations and eliminating the derivatives of u leads to a new set of equations (D R )u = 0 D F F ψ F + F ψ

45 Discussion Proceeding in this way we get a sequence of sets of equations R u = 0, (D R )u = 0, (D ll D R )u = 0, If p is the number of independent equations in the first K sets, then the general solution depends on N p arbitrary constants.

46 Review II: Prolongation of PDEs

47 Prolongation F x, f, f, f, = 0 u x = ψ u i = 1,, n α = 1,, N

48 Example 1 Introduce w = u v u = au + bv u + v = cu + dv v = eu + fv u = au + bv u = 1 (cu + dv + w) 2 v = 1 (cu + dv w) 2 v = eu + fv w = w (u, v, w) w = w (u, v, w)

49 Example 2: Cauchy-Riemann equation u = v u = v Impossible to make a prolongation! In fact, solution of this system depends on one holomorphic function.

50 Prolongation F x, f, f, f, = 0 Not always possible When can we make a prolongation successfully? u x = ψ (x, u) i = 1,, n α = 1,, N

51 Prolongation of Killing equations

52 Spacetime symmetry Killing vector fields: ξ μ ξ ν + ν ξ μ = 0 Killing-Stackel tensors (μ K ν1 ν 2 ν n ) = 0 Killing-Yano tensors (μ f ν1 )ν 2 ν n = 0 [Stackel 1895] [Yano 1952] K (μ1 μ 2 μ n ) = K μ1 μ 2 μ n f [μ1 μ 2 μ n] = f μ1 μ 2 μ n

53 Killing vectors μ ξ ν + ν ξ μ = 0

54 Killing vector equation μ ξ ν + ν ξ μ = 0 μ ξ ν = L μν, L μν = [μ ξ ν] μ L νρ = R νρμ σ ξ σ

55 μ ξ ν = L μν, L μν = L μν μ L νρ = R νρμ σ ξ σ μ ξ ν L νρ 0 R σ νρμ 1 0 ξ σ L μν = 0 D μ ξ A = 0 ξ A = (ξ μ, L μν ) : a section of Λ 1 M Λ 2 M D μ : connection on Λ 1 M Λ 2 M ξ ν 0 1 D μ ξ A μ L νρ R σ νρμ 0 ξ σ L μν

56 Point 1 Prolongation Killing vectors parallel sections of Λ 1 M Λ 2 M ξ μ ξ A = ξ μ [μ ξ ν] s.t. D μ ξ A = 0 Parallel equation The number of linearly independent sections of Λ M Λ M is bound by the rank of the vector bundle. N = n 1 + n 2 = n n + 1 2

57 Maximally symmetric spaces Spaces that have the maximum number of KVs constant curvature spaces n N = n n

58 Point ➁ Curvature conditions # of parallel sections = rank of Ep - # of curv. cond. [D μ, D ν ] ξ A = 0 D μ ξ A = 0 D μ, D ν, D ρ ξ A = 0 D μ, D ν, D ρ, D σ ξ A = 0

59 Killing-Yano tensors (μ f ν1 )ν 2 ν n = 0 f [μ1 μ 2 μ n] = f μ1 μ 2 μ n

60 KY tensor equation (μ f ν)ρ = 0 f μν = f νμ μ f νρ = [μ f νρ] μ ( [ν f ρσ] ) = R [νρ μ α f α σ]

61 Rank-2 μ f νρ = [μ f νρ] μ ( [ν f ρσ] ) = R [νρ μ α f α σ] Rank-p μ f ν1 ν p = [μ f ν1 ν p ] μ ( [ν f ρ1 ρ p ]) = R [νρ1 μ α f α ρ2 ρ p ]

62 Prolongation of KY tensors rank-p KY tensors parallel sections of E p E = Λ M Λ (M) = p p + 1 rank(e ) = n + 1 p + 1

63 The number of KY tensors in maximally symmetric spaces N = n + 1 p + 1 Semmelmann 2002 p=1 p=2 p=3 p=4 2D 3 3D 6 4 4D D

64 Examples in four dimensions TH-Yasui D metrics p = 1 p = 2 p = 3 Maximally symmetric Plebanski-Demianski Kerr Schwazschild FLRW Self-dual Taub-NUT Eguchi-Hanson 4 3 0

65 Examples in five dimensions TH-Yasui D metrics Maximally symmetric Myers-Perry Emparan-Reall Kerr string p = 1 p = 2 p = p =

66 Killing-Stackel tensors (μ K ν1 ν 2 ν n ) = 0 K (μ1 μ 2 μ n ) = K μ1 μ 2 μ n

67 (μ K νρ) = 0 K μν = K νμ μ K νρ = 2 3 [μk ν]ρ + [μ K ρ]ν μ [ν K ρ]σ = R νρ(μ α K α σ) R (μ [νρ] α K α σ) 1 4 R νρ[μ α K α σ 1 2 R (μ [νρ] α K α σ + φ [μ νρ σ] where φ μνρσ (μ ν) K ρσ μ φ [ν ρσ κ] = R 1 K μνρσκ + R 2 [ K ] μνρσκ

68 Prolongation of KS tensors rank-p KS tensors parallel sections of E p p E = rank(e ) = 1 n n + p p + 1 n + p 1 p

69 The number of KS tensors in maximally symmetric spaces N = 1 n n + p p + 1 n + p 1 p Barbance 1973, Michel et al 2012 p=1 p=2 p=3 p=4 2D D D D

70 On-going tasks Analysis of curvature conditions - Compute the curvature conditions - Construct the package of Mathematica which compute and solve the curvature conditions - Investigate the curvature conditions for various metrics Conjecture No non-trivial quadratic constant for geodesic motion in the Kerr spacetime exists, with the exception of Carter constant.

71 Foresight into the future CKY and CKS Cotton tensor, Bach tensor, Q-curvature, conformal geometry PDE theory Prolongation Differential geometry Generalised gradients, Weitzenbock formula, twisted Dirac Hamiltonian dynamics Integrable systems, Chaos, Lax pairs, Painleve systems GR, SUGRA, Exact solutions, strings, branes