Eva Miranda. UPC-Barcelona and BGSMath. XXV International Fall Workshop on Geometry and Physics Madrid

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1 b-symplectic manifolds: going to infinity and coming back Eva Miranda UPC-Barcelona and BGSMath XXV International Fall Workshop on Geometry and Physics Madrid Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

2 Outline 1 Motivating examples 2 b n -Symplectic manifolds 3 Integrable systems on b-symplectic manifolds 4 Deblogging b n -symplectic manifolds Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

3 The restricted 3-body problem (joint with Delshams) Simplified version of the general 3-body problem. One of the bodies has negligible mass. The other two bodies move independently of it following Kepler s laws for the 2-body problem on elliptic orbits with eccentricity e. Figure: Circular 3-body problem Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

4 Planar restricted 3-body problem By Newton s law of universal gravitation d 2 q dt 2 = (1 µ) q 1 q q 1 q 3 + µ q 2 q q 2 q 3 (1) with q 1 = q 1 (t) the position of the planet with mass 1 µ at time t and q 2 = q 2 (t) the position of the one with mass µ. Introducing the momentum p = dq/dt and the time-dependent self-potential of the small body U(q, t) = 1 µ q q + µ 1 q q 2 Equation (1) can be rewritten as a Hamiltonian system with Hamiltonian H(q, p, t) = p 2 /2 U(q, t), (q, p) R 2 R 2,. The primaries move around their center of mass on ellipses and it is useful to introduce polar coordinates for q = (X, Y ) R 2 \ {0}, X = r cos α, Y = r sin α, (r, α) R + T The momenta p = (P X, P Y ) are transformed in such a way that the total change of coordinates (X, Y, P X, P Y ) (r, α, P r =: y, P α =: G) is canonical, i.e. the symplectic structure remains the same. Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

5 Planar restricted 3-body problem To study the behaviour at r =, we introduce McGehee coordinates (x, α, y, G), where r = 2 x 2, x R+. This transformation is non-canonical i.e. the symplectic structure changes: for x > 0 it is given by 4 dx dy + dα dg. x3 which extends to a b 3 -symplectic structure on R T R 2. The Poisson bracket is {f, g} = x3 4 ( f g g y f ) g + f y x α g G f g G α The integrable 2-body problem for µ = 0 is integrable with respect to the singular ω. Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

6 Model for these systems ω = 1 x1 n dx 1 dy 1 + dx i dy i i 2 Close to x 1 = 0, the systems behave like, and not like, Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

7 The 3-body problem Consider the system of three bodies with masses m 1, m 2, m 3 and positions q 1 = (q 1, q 2, q 3 ), q 2 = (q 4, q 5, q 6 ), q 3 = (q 7, q 8, q 9 ) R 3. Define the 9 9 matrix M := diag(m 1, m 1, m 1, m 2, m 2, m 2, m 3, m 3, m 3 ). Assume central coordinates (m 1 q 1 + m 2 q 2 + m 3 q 3 = 0). Introduce the following McGehee -coordinates: r := q T Mq, s := q r, z := p r. (2) r = 0 corresponds to triple collisions. Essentially, these are spherical coordinates since s lies on the unit-sphere in R 9 with respect to the metric given by M. Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

8 The 3-body problem The standard symplectic form 9 i=1 dq i dp i becomes in the new coordinates (r, s 1,..., s 8, z 1,..., z 9 ). 8 i=1 ( si dr dz i + rds i dz i z ) i r 2 r ds i dr m9 rµ ( µdr dz 9 r with µ := 1 8 i=1 s2 i m i. 9 dq i dp i = i=1 µr 7 8 i=1 m i s i ds i dz 9 + z 9 2 ) 8 m i s i ds i dr, i=1 m 9 ds 1 dz 1 ds 2 dz 2... ds 8 dz 8 dr dz 9, It is a 7 2-folded symplectic structure. (In the n-body problem m-folded symplectic for a certain m). Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

9 Other examples Kustaanheimo-Stiefel regularization for n-body problem (useful for binary collisions) folded-type symplectic structures with hyperbolic singularities two fixed-center problem via Appell s transformation combination of folded-type and b m -symplectic structures Dirac structures. Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

10 b-symplectic/b-poisson structures Definition Let (M 2n, Π) be an (oriented) Poisson manifold such that the map p M (Π(p)) n Λ 2n (T M) is transverse to the zero section, then Z = {p M (Π(p)) n = 0} is a hypersurface called the critical hypersurface and we say that Π is a Poisson b-structure on (M, Z). Symplectic foliation of a b-poisson manifold The symplectic foliation has dense symplectic leaves and codimension 2 symplectic leaves whose union is Z. Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

11 Darboux normal forms Theorem (Guillemin-M.-Pires) For all p Z, there exists a Darboux coordinate system x 1, y 1,..., x n, y n centered at p such that Z is defined by x 1 = 0 and Π = x 1 + x 1 y 1 n i=2 x i y i Darboux for b n -symplectic structures or dually Π = x n 1 + x 1 y 1 ω = 1 x n dx 1 dy n i=2 x i y i n dx i dy i i=2 Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

12 Dimension 2 Radko classified b-poisson structures on compact oriented surfaces: Geometrical invariants: The topology of S and the curves γ i where Π vanishes. Dynamical invariants: The periods of the modular vector field along γ i. Measure: The regularized Liouville volume of S, V ε h (Π) = h >ε ω Π for h a function vanishing linearly on the curves γ 1,..., γ n and ω Π the dual form to the Poisson structure. Other classification schemes: For b n -symplectic structures (not necessarily oriented) Scott, M.-Planas. Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

13 Higher dimensions: Some compact examples. The product of (R, π R ) a Radko compact surface with a compact symplectic manifold (S, ω) is a b-poisson manifold. corank 1 Poisson manifold (N, π) and X Poisson vector field (S 1 N, f(θ) θ X + π) is a b-poisson manifold if, 1 f vanishes linearly. 2 X is transverse to the symplectic leaves of N. We then have as many copies of N as zeroes of f. Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

14 Poisson Geometry of the critical hypersurface This last example is semilocally the canonical picture of a b-poisson structure. 1 The critical hypersurface Z has an induced regular Poisson structure of corank 1. 2 There exists a Poisson vector field transverse to the symplectic foliation induced on Z. Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

15 The singular hypersurface Theorem (Guillemin-M.-Pires) If L contains a compact leaf L, then Z is the mapping torus of the symplectomorphism φ : L L determined by the flow of a Poisson vector field v transverse to the symplectic foliation. This description also works for b n -symplectic structures. Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

16 A dual approach... b-poisson structures can be seen as symplectic structures modeled over a Lie algebroid (the b-tangent bundle). A vector field v is a b-vector field if v p T p Z for all p Z. The b-tangent bundle b T M is defined by { } Γ(U, b b-vector fields T M) = on (U, U Z) Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

17 b-forms The b-cotangent bundle b T M is ( b T M). Sections of Λ p ( b T M) are b-forms, b Ω p (M).The standard differential extends to d : b Ω p (M) b Ω p+1 (M) A b-symplectic form is a closed, nondegenerate, b-form of degree 2. This dual point of view, allows to prove a b-darboux theorem and semilocal forms via an adaptation of Moser s path method because we can play the same tricks as in the symplectic case. What else? Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

18 b-integrable systems Definition b-integrable system A set of b-functions a f 1,..., f n on (M 2n, ω) such that f 1,..., f n Poisson commute. df 1 df n 0 as a section of Λ n ( b T (M)) on a dense subset of M and on a dense subset of Z a c log x + g Example The symplectic form 1 hdh dθ defined on the interior of the upper hemisphere H + of S 2 extends to a b-symplectic form ω on the double of H + which is S 2. The triple (S 2, ω, log h ) is a b-integrable system. Example If (f 1,..., f n ) is an integrable system on M, then (log h, f 1,..., f n ) on H + M extends to a b-integrable on S 2 M. Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

19 Action-angle coordinates for b-integrable systems The compact regular level sets of a b-integrable system are (Liouville) tori. Theorem (Kiesenhofer-M.-Scott) Around a Liouville torus there exist coordinates (p 1,..., p n, θ 1,..., θ n ) : U B n T n such that ω U = c p 1 dp 1 dθ 1 + n dp i dθ i, (3) i=2 and the level sets of the coordinates p 1,..., p n correspond to the Liouville tori of the system. Reformulation of the result Integrable systems semilocally twisted cotangent lift a of a T n action by translations on itself to (T T n ). a We replace the Liouville form by c log p 1 dθ 1 + n i=2 pidθi. Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

20 Intermezzo on twisted b-cotangent lifts Consider G := S 1 R + S 1 acting on M := S 1 R 2 : (ϕ, a, α) (θ, x 1, x 2 ) := (θ + ϕ, ar α (x 1, x 2 )), with R α rotation. Its twisted b-cotangent lift gives focus-focus singularities on b-symplectic manifolds. The logarithmic Liouville one-form is λ := log p dθ + y 1 dx 1 + y 2 dx 2 and the moment map is µ := (f 1, f 2, f 3 ) with f 1 = λ, X # 1 = log p, f 2 = λ, X # 2 = x 1y 1 + x 2 y 2, f 3 = λ, X # 3 = x 1y 2 y 1 x 2. Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

21 Proof 1 Topology of the foliation. In a neighbourhood of a compact connected fiber the b-integrable system F is diffeomorphic to the b-integrable system on W := T n B n given by the projections p 1,..., p n 1 and log p n. 2 Uniformization of periods: We want to define integrals whose (b-)hamiltonian vector fields induce a T n action. Start with R n -action: Φ : R n (T n B n ) T n B n ((t 1,..., t n ), m) Φ (1) t 1 Φ (n) t n (m). Uniformize to get a T n action with fundamental vector fields Y i. 3 The vector fields Y i are Poisson vector fields (check L Yi L Yi ω = 0). 4 The vector fields Y i are Hamiltonian with primitives σ 1,..., σ n b C (W ). In this step the properties of b-cohomology are essential.use this action to drag a local normal form (Darboux-Carathéodory) in a whole neighbourhood. Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

22 Proof 1 Topology of the foliation. In a neighbourhood of a compact connected fiber the b-integrable system F is diffeomorphic to the b-integrable system on W := T n B n given by the projections p 1,..., p n 1 and log p n. 2 Uniformization of periods: We want to define integrals whose (b-)hamiltonian vector fields induce a T n action. Start with R n -action: Φ : R n (T n B n ) T n B n ((t 1,..., t n ), m) Φ (1) t 1 Φ (n) t n (m). Uniformize to get a T n action with fundamental vector fields Y i. 3 The vector fields Y i are Poisson vector fields (check L Yi L Yi ω = 0). 4 The vector fields Y i are Hamiltonian with primitives σ 1,..., σ n b C (W ). In this step the properties of b-cohomology are essential.use this action to drag a local normal form (Darboux-Carathéodory) in a whole neighbourhood. Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

23 Proof 1 Topology of the foliation. In a neighbourhood of a compact connected fiber the b-integrable system F is diffeomorphic to the b-integrable system on W := T n B n given by the projections p 1,..., p n 1 and log p n. 2 Uniformization of periods: We want to define integrals whose (b-)hamiltonian vector fields induce a T n action. Start with R n -action: Φ : R n (T n B n ) T n B n ((t 1,..., t n ), m) Φ (1) t 1 Φ (n) t n (m). Uniformize to get a T n action with fundamental vector fields Y i. 3 The vector fields Y i are Poisson vector fields (check L Yi L Yi ω = 0). 4 The vector fields Y i are Hamiltonian with primitives σ 1,..., σ n b C (W ). In this step the properties of b-cohomology are essential.use this action to drag a local normal form (Darboux-Carathéodory) in a whole neighbourhood. Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

24 Proof 1 Topology of the foliation. In a neighbourhood of a compact connected fiber the b-integrable system F is diffeomorphic to the b-integrable system on W := T n B n given by the projections p 1,..., p n 1 and log p n. 2 Uniformization of periods: We want to define integrals whose (b-)hamiltonian vector fields induce a T n action. Start with R n -action: Φ : R n (T n B n ) T n B n ((t 1,..., t n ), m) Φ (1) t 1 Φ (n) t n (m). Uniformize to get a T n action with fundamental vector fields Y i. 3 The vector fields Y i are Poisson vector fields (check L Yi L Yi ω = 0). 4 The vector fields Y i are Hamiltonian with primitives σ 1,..., σ n b C (W ). In this step the properties of b-cohomology are essential.use this action to drag a local normal form (Darboux-Carathéodory) in a whole neighbourhood. Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

25 A picture... Figure: Fibration by Liouville tori Applications to KAM theory (surviving torus under perturbations ) on b-symplectic manifolds (Kiesenhofer-M.-Scott). Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

26 KAM for b-symplectic manifolds Theorem (Kiesenhofer-M.-Scott) Consider T n Br n with the standard b-symplectic structure and consider the b-function H = k log y 1 + h(y) with h analytic. If the frequency map has a Diophantine value and is non-degenerate, then a Liouville torus on Z persists under sufficiently small perturbations of H. More precisely, if ɛ is sufficiently small, then the perturbed system H ɛ = H + ɛp (with P (ϕ, y) = log y 1 + f 1 ( ϕ, y) + y 1 f 2 (ϕ, y) + f 3 (ϕ 1, y 1 )) admits an invariant torus T. Moreover, there exists a diffeomorphism T n T close to the identity taking the flow γ t of the perturbed system on T to the linear flow on T n with frequency vector ( k+ɛk c, ω). Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

27 Global classification Circle actions on b-surfaces: m 1 m 3 m 5 m 1 m 2 m 3 m 4 m 2 m 4 Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

28 Further results on b-manifolds Delzant theorem and convexity for T k -actions (Guillemin- M.-Pires-Scott). Symplectic topological aspects (Frejlich-Martínez-M). Quantization of b-symplectic manifolds (Guillemin- M.-Weitsman). What about b n -symplectic manifolds? Guillemin-M.-Weitsman b n -symplectic Symplectic Folded symplectic Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

29 Desingularizing b m -symplectic structures Theorem (Guillemin-M.-Weitsman) Given a b m -symplectic structure ω on a compact manifold (M 2n, Z): If m = 2k, there exists a family of symplectic forms ω ɛ which coincide with the b m -symplectic form ω outside an ɛ-neighbourhood of Z and for which the family of bivector fields (ω ɛ ) 1 converges in the C 2k 1 -topology to the Poisson structure ω 1 as ɛ 0. If m = 2k + 1, there exists a family of folded symplectic forms ω ɛ which coincide with the b m -symplectic form ω outside an ɛ-neighbourhood of Z. Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

30 0 Deblogging b 2k -symplectic structures ω = dx 2k 1 x 2k ( α i x i ) + β (4) Let f C (R) be an odd smooth function satisfying f (x) > 0 for all x [ 1, 1], i=0-1 1 and satisfying f(x) = { 1 2 (2k 1)x 2k 1 for x < (2k 1)x 2k 1 for x > 1 outside [ 1, 1]. Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

31 Deblogging b 2k -symplectic structures Scaling: f ɛ (x) := 1 ( x ) ɛ 2k 1 f. (5) ɛ Outside the interval [ ɛ, ɛ], Replace dx x 2k f ɛ (x) = { by df ɛ to obtain 1 2 (2k 1)x 2k 1 ɛ 2k 1 for x < ɛ (2k 1)x 2k 1 ɛ 2k 1 for x > ɛ which is symplectic. 2k 1 ω ɛ = df ɛ ( α i x i ) + β i=0 Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27

32 Applications of deblogging Convexity for T k -actions. Delzant theorem and Delzant-type theorem for semitoric systems (bolytopes). Applications to KAM. Periodic orbits of problems in celestial mechanics and applications to stability (joint with Roisin Braddell, Amadeu Delshams, Cédric Oms and Arnau Planas). Eva Miranda (UPC) b-symplectic manifolds Semptember, / 27