Monodromy stories: relation to integrability, numerics, applications. Vladimir Salnikov INSA de Rouen Universit e de Caen GRAVASCO trimester Paris, 26 September 2013.
Plan Ziglin's method for analysis of integrability Monodromy group Ecient algorithm of application of the Ziglin's method Examples: - Double pendulum - HenonHeiles system - Satellite dynamics - 3-body problem - Bianchi universes Bonus: Rigid body dynamics
Integrability LiouvilleArnold theorem (complete integrability): Let n smooth functions F i on a 2n-dimensional manifold M be in involution: {F i, F j } = 0. Consider the level set of functions F i M f = {(q, p) : F i (q, p) = f i, i = 1,..., n}. Let the functions F i be independent on M f. Then: 1. M f is invariant under the phase ow with the hamiltonian function H = F 1 (q, p) (ṗ i = H q i, q i = H ). p i 2. If M f is compact, then each connected component of it is dieomorphic to an n-dimensional torus T n = {(ϕ 1,..., ϕ n ) mod 2π} 3. The phase ow with the hamiltonian H denes on M f a quasi-periodic motion: ϕ = ω(f) 4. Canonical hamiltonian equations can be integrated in quadratures. In this case the system is called completely integrable or integrable in LiouvilleArnold sense
Meromorphic integrability Two mathematicians talking: I lost the key from the door, how do I get back home?.. Is it a real door? Sure, why? Go to the complex space, go around it, then come back.
Variational equations Consider in C 2n a system ẋ = v(x) (1) with complex time. Fix some particular solution x 0 (t) of (1). Plug x = x 0 + ξ to (1) and obtain ξ = A(t)ξ +..., A = v x (x 0(t)) Then the linearized system ξ = A(x 0 (t))ξ is called a system of variational equations along a particular solution x 0 ( ).
Theorem 1. (Ziglin's Lemma) Let the monodromy group M of a curve x 0 contain a non-resonant transformation. Then the number of meromorphic rst integrals of hamiltonian equations in the connected neighbourhood of x 0 functionally independent with H is less than or equal to the number of rational invariants of M. Theorem 2. (Ziglin) Let the monodromy group M of a curve x 0 contain a non-resonant transformation g. Then for the hamiltonian equations to have (n 1) meromorphic rst integrals in a connected neighbourhood of x 0, functionally independent with H, it is necessary that, any transformation g M preserves the xed point of g and maps all the eigendirections of g to eigendirections. If moreover no set of eigenvalues of g forms a regular polygon centered in 0 on a complex plane, (with number of vertices 2), then g preserves eigendirections of g (i.e. commutes with g ).
Monodromy group (simple algebraic example) Consider the equation ξ 5 + ξx + 1 = 0 The solution of ξ = ξ(x) 5-valued function. Change x along a closed loop (avoiding multiple roots) possible permutation of roots. Group structure: product = concatenation of loops, inverse = following the loop in the opposite direction. Monodromy group M = all permutations of 5 elements.
Monodromy group Initial system: ẋ = v(x) with a particular solution x 0 (t) Riemann surface. Variational equations along x 0 : ξ = A(x0 (t))ξ M γ : (ξ 1,..., ξ n ) ( ξ 1,..., ξ n ), M = {M γ, γ π 1 (x 0 )}
Properties Elements of the monodromy group are symplectic ane transformations eigenvalues go in couples: λ, λ 1. Monodromy group can be viewed as a set of dierential automorphisms of some eld preserving the coecients of the linearized system. (Exercise) What is the relation to the Kovalevskaya matrix/exponents and the Yoshida's method?
Ziglin's method Standard application procedure: 1. For a given (initial) complexied system of dierential equations construct an explicit particular solution. 2. Write down the system of variational equations. If possible reduce the order of this system to normal variational equations. 3. Find the singular points of the particular solution. (Not the branching points of the particular solution of time, but singularities on a Riemann surface). 4. Construct the monodromy matrices, corresponding to the loops around these singularities they will generate the monodromy group. 5. Having computed the commutators of these matrices conclude on the possible integrability.
Diculties Standard application procedure: 1. For a given (initial) complexied system of dierential equations construct an explicit particular solution. 2. Write down the system of variational equations. If possible reduce the order of this system to normal variational equations. 3. Find the singular points of the particular solution. (Not the branching points of the particular solution of time, but singularities on a Riemann surface). 4. Construct the monodromy matrices, corresponding to the loops around these singularities they will generate the monodromy group. 5. Having computed the commutators of these matrices conclude on the possible integrability.
Ways out Richer group (e.g. MoralesRamis method via dierential Galois theory = maximal group satisfying the property above)
MoralesRamis Lemma. If the system of equation possesses a meromorphic rst integral in the neighbourhood of a curve x 0 (particular solution), then the dierential Galois group G of the system of normal variational equations possesses a rational rst integral. Theorem (MoralesRamis) Let the hamiltonian system be LiouvilleArnold integrable in the neighbourhood of its particular solution. Then the identity component of the dierential Galois group of the corresponding system of normal variational equations is abelian. Remark 1. M G, but construction of G is less explicit. Remark 2. Kovacic algorithm for G (second order equation).
Ways out Richer group (e.g. MoralesRamis method via dierential Galois theory = maximal group satisfying the property above) Higher variations (ZiglinMoralesRamisSimo theory) Numerical methods: Roekaerts D, Yoshida H, J. Phys. A: Math. Gen. 21, 3547-3557, 1988. Maciejewski A J, God ziewski K, Reports on Math. Phys. 44, 133-142, 1999. C. Sim o. Computer assisted studies in dynamical systems. In: Progress in nonlinear science. Vol. I:Mathematical problems of nonlinear dynamics (Eds.: L. M. Lerman, L. P. Shilnikov). Univ. Nizhny Novgorod Press, 2002, p. 152-165. Remark. Only the second diculty...
Diculties Standard application procedure: 1. For a given (initial) complexied system of dierential equations construct an explicit particular solution. 2. Write down the system of variational equations. If possible reduce the order of this system to normal variational equations. 3. Find the singular points of the particular solution. (Not the branching points of the particular solution of time, but singularities on a Riemann surface). 4. Construct the monodromy matrices, corresponding to the loops around these singularities they will generate the monodromy group. 5. Having computed the commutators of these matrices conclude on the possible integrability.
Eective algorithm of application of the Ziglin's method (1) 1) Write down an extended system of equations (not xing any particular solution) ẋ = v(x) Ξ = A(x)Ξ, (2) where the rst line is the initial complexied system, second matrix equation, where A is the matrix of the system of variational equations, depending explicitly on x, Ξ unknown 2n 2n matrix (for 2n dimension of the phase space of the initial system). 2) Choose a domain C and a grid Gr of points in it with some distinguished point t 0.
Monodromy group Initial system: ẋ = v(x) Variational equations: Ξ = A(x)Ξ M γ : (ξ 1,..., ξ n ) ( ξ 1,..., ξ n ), M = {M γ, γ π 1 (x 0 )}
Eective algorithm of application of the Ziglin's method (2) 3) For each point from Gr choose a closed loop, going around it starting from t 0. Integrate the system (2) along this loop for the initial data Ξ(t 0 ) = id. Three options are possible: i. x and Ξ returned to initial values this point gives trivial transformation from the monodromy group. ii. x didn't return to the initial value (within a given precision) repeat integration along the same loop. If x didn't return to the initial value after several iterations, analize the density of this trajectory in the phase space. iii. Values of x returned to the initial data, but Ξ not store the obtained matrix Ξ, it will be one of the generators of the monodromy group. 4) Compute pairwise commutators of the matrices from 3)iii. If there are non-zero commutators, conclude meromorphic non-integrability. If not, choose another point t 0 in 2) or initial value x(t 0 ) in 3).
Eective algorithm steps 1-2
Eective algorithm step 3
Eective algorithm step 3
Eective algorithm step 3 outcome 3.i or 3.iii
Eective algorithm step 4
Implementation details Controllable precision Inequation condition Independent trajectories Parallelization with linear gain (Eg. CUDA for GPU) V.S.: Eective algorithm of analysis of integrability via the Ziglin's method, arxiv:1208.6252 [math.ds]; V.S.: Integrability of the double pendulum the Ramis' question, arxiv:1303.4904 [math.ds]; V.S.: Int egrabilit e dynamique : de l'approche alg ebrique au calcul parall ele, Matapli, SMAI, N 100, 2013.
Double pendulum Ramis' question 2 1.5 γ 1 g 1 0.5 t 1 α 1 Im(t) 0 t 0-0.5-1 t 2 α 2-1.5 γ 2-2 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Re(t) t 1 = 0.5 + 0.9i (3 times) t 2 = 0.5 0.9i (3 times) 20.72 17.12i 15.79 1.34i 4.94 + 29.46i 14.55 + 10.90i 17.67 + 12.78i 12.24 0.06i 1.93 24.87i 12.91 7.99i 12.28 + 6.91i 3.55 + 7.78i 13.07 + 7.86i 8.18 5.42i 11.84 12.56i 1.31 10.39i 19.32 4.06i 8.59 + 9.31i 18.72 17.12i 15.79 1.34i 4.94 + 29.46i 14.55 + 10.90i 17.67 + 12.78i 14.24 0.06i 1.93 24.87i 12.91 7.99i 12.28 + 6.91i 3.55 + 7.78i 15.07 + 7.86i 8.18 5.42i 11.84 12.56i 1.31 10.39i 19.32 4.06i 6.59 + 9.31i Don't commute meromorphic non-integrability.
Application: HenonHeiles system H = 1 2 (p2 1 + p 2 2) q 2 2(A + q 1 ) λ 3 q3 1 For λ 0 not all the cases can be studied within the approach of MoralesRamis we apply the algorithm above. Example: for λ = 1, A = 0.25 and initial data (q 1, q 2, p 1, p 2 ) = (1, 0.4, 1.25, 0.3), t 0 = 1. The monodromy matrices corresponding to the points (0.2 + 2.5i) and (0.2 2.5i) have the commutator 0.390 0.913i 0.658 3.463i 1.099 + 2.566i 0.655 2.223i 0.937 + 2.301i 0.637 + 2.599i 1.311 1.866i 1.720 + 7.996i 0.315 0.760i 0.274 1.212i 0.520 + 0.878i 0.576 2.514i 0.464 1.079i 0.797 4.199i 1.335 + 3.133i 0.765 2.564i that is meromorphic non-integrability.,
Application: Satellite dynamics Axially symmetrical rigid body on a circular orbit. H = p2 ψ 2 θ 2 sin 2 θ + p 2 p ψctg(θ) cos(ψ) cos θ αβpφ sin 2 θ p θ sin(ψ) + αβ cos ψ sin θ + α2 β 2 2 sin 2 θ + 3 (α 1) 2 cos2 θ, for α = C/A; A, B, C principal moments of inertia (A = B). The ϕ coordinate is cyclic, so x p ϕ = αβ = const, consider the values α = 4/3, β = 0 (cf. B. Bardin et al). For a xed value of p ϕ the Routh reduced system has 2 degrees of freedom and is described by a hamiltonian: H = p2 ψ 2 sin 2 θ + p 2 θ 2 p ψ + 1 2 sin2 ψ sin 2 θ.
Application: Satellite dynamics 2 θ H = p2 ψ 2 sin 2 θ + p 2 p ψ + 1 2 sin2 ψ sin 2 θ. Take the trajectory starting at ψ = 0, θ = 1, p ψ = 0.1, p θ = 0, t = 0 and go around t = 4.8 + 0.8i and t = 4.8 0.8i (twice). The corresponding commutator is: 8849.8 + 13.2915i 37.9467 126.071i 2044.45 + 35.4031i 1843.31 125.866i 9456.28 239.498i 311.834 62.6925i 2350.67 37.0663i 1972.34 + 197.277i 34540.9 527.522i 596.362 53.1627i 8340.91 82.1358i 7205.45 + 624.613i 4032.64 556.427i 1615.13 971.49i 177.875 + 135.104i 820.725 + 131.537i that is meromorphic non-integrability.
Application: planar 3-body problem H = 3 i=1 p i 2 2m i 1 j<k 3 m j m k q j q k
Reduced system Using Poincar e (1896) reduction: H = M 1 2 ( p 2 1 + 1 q 2 1 ) ( P 2 + M 1 ( ) p 2 2 + p 2 1 3 + p 1 p 2 p 3 P 2 m 3 q 1 m 1 m 2 m 1m 3 m 2m 3 q 1 q2 2 + q2 3 ) (q 1 q 2 ) 2 + q 2 3 for P = p 3 q 2 p 2 q 3 k, M 1 = 1 m 3 + 1 m 1, M 2 = 1 m 3 + 1 m 2. Integrability analysis: A. Tsygvintsev '01, D. Boucher and J.-A. Weil '01. Consider Lagrangian solution = equilateral triangle compute the monodromy/dierential Galois group.
Lagrangian solution details q 1 distance between two bodies, q 2, q 3 projections of the third body to two orthogonal directions, p i respective momenta. H = 0 q 2, q 3, p i functions of q 1. The parametrization is via w = p 1 q 1 C internal time (is OK for analytics if one is able to write variational equations). Remark 1. To relate to dynamics need external time: αw + β (2aw + b)ẇ = aw 2 + bw + d One real, two complex singularities. Important: should not create/remove essential singularities by reparametrization! Remark 2. Some parameter values when VEs have invariants (Tsygvintsev '06). Remark 3. What if we modify the interaction law? U(r 1,..., r n ) := m i m j r i (t) r j (t r i r j /c) i,j
Application: Bianchi universe(s) Joint project with Guillame Duval and J er ome Perez: Salvador Dali jewels: Big Bang
Bianchi IX universe Ȧ 1 Ȧ 2 + Ȧ 2 Ȧ 3 + Ȧ 3 Ȧ 1 + 2e A 1+A 2 + 2e A 2+A 3 + 2e A 3+A 1 e 2A 1 e 2A 2 e 2A 3 4J 0 e P 0 2 (A 1 +A 2 +A 3 ) = 0 Ä 1 = J 0 P 0 e P 0 2 (A 1 +A 2 +A 3 ) e 2A 1 + e 2A 2 + e 2A 3 2e A 2+A 3 Ä 2 = J 0 P 0 e P 0 2 (A 1 +A 2 +A 3 ) + e 2A 1 e 2A 2 + e 2A 3 2e A 1+A 3 Ä 3 = J 0 P 0 e P 0 2 (A 1 +A 2 +A 3 ) + e 2A 1 + e 2A 2 e 2A 3 2e A 1+A 2 Facts: The system is hamiltonian Empty universe generically non-integrable (J. Morales '99) H = 0? (cf. A. Maciejewski et al. '01 ) Relation to billiards
Numerics for Bianchi IX Empty universe case (very symmetric solution): Gradient norm for complex time. Monodromy matrices for these loops do not commute
Numerics for Bianchi IX Non-empty universe case (arbitrary solutions):
Numerics for Bianchi IX Non-empty universe case (arbitrary solutions): Analytics in progress.
BONUS
KAM theory for rigid body dynamics perturbation Theorem of KolmogorovArnoldMoser: For a system obtained from an integrable one by a small perturbation, the probability to stay on an invariant torus, preserving its topology, decreases when the perturbation parameter increases. Initial conditions BLACK BOX Invariant torus or not V.S.: On numerical approaches to the analysis of topology of the phase space for dynamical integrability arxiv:1206.3801v1 [math.ds]; We reconstruct the distribution of preserved invariant tori by the Monte-Carlo method.
KAM theory for rigid body dynamics In the body-xed frame: EulerPoisson equations J ω + ω Jω = mgr G γ, γ + ω γ = 0, γ is the direction of the gravity force, ω angular velocity. Parameters: m mass, g gravity constant, rg position of the center of mass, J matrix of inertia with the eigenvalues A, B, C. First integrals: H 1 2 Jω ω + mgr G γ, K Jω γ, l 2 γ 2. Hamiltonian on the level set M = {(ω, γ), K = k, l 2 = 1}
Generalized method of sections 1. Arbitrary A, B, C ; rg = (xg, yg, zg ) varying around 0. Proportion of empty sections depending on xg (yg = zg = 0). Euler case: center of mass is xed. J ω 2 = const
KAM theory for rigid body dynamics 2. Symmetric body: A = B = 2, rg in their eigenplane. Proportion of empty sections depending on C. 2.a) Kovalevskaya case: A = B = 2C. 2.b) Lagrange case: r parallel to the symmetry axis.
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