00899nas a2200109   4500008004300000245006800043210006800111520053400179100002000713700002000733856003600753       2007                        en_Ud  00aCanonical structure and symmetries of the Schlesinger equations0 aCanonical structure and symmetries of the Schlesinger equations3 aThe Schlesinger equations S (n,m) describe monodromy preserving deformations of order m Fuchsian systems with n+1 poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of n copies of m×m matrix algebras equipped with the standard linear Poisson bracket. In this paper we present a new canonical Hamiltonian formulation ofthe general Schlesinger equations S (n,m) for all n, m and we compute the action of the symmetries of the Schlesinger equations in these coordinates.1 aDubrovin, Boris1 aMazzocco, Marta  uhttp://hdl.handle.net/1963/199700931nas a2200121   4500008004100000245007500041210006800116260001000184520053900194100002000733700002000753856003600773       2007                        en  d00aOn the reductions and classical solutions of the Schlesinger equations0 areductions and classical solutions of the Schlesinger equations  bSISSA3 aThe Schlesinger equations S(n,m) describe monodromy preserving\\r\\ndeformations of order m Fuchsian systems with n+1 poles. They\\r\\ncan be considered as a family of commuting time-dependent Hamiltonian\\r\\nsystems on the direct product of n copies of m×m matrix algebras\\r\\nequipped with the standard linear Poisson bracket. In this paper we address\\r\\nthe problem of reduction of particular solutions of “more complicated”\\r\\nSchlesinger equations S(n,m) to “simpler” S(n′,m′) having n′ < n\\r\\nor m′ < m.1 aDubrovin, Boris1 aMazzocco, Marta  uhttp://hdl.handle.net/1963/647201274nas a2200109   4500008004300000245006000043210006000103260001300163520093200176100002001108856003601128       2001                        en_Ud  00aPicard and Chazy solutions to the Painlevé VI equation0 aPicard and Chazy solutions to the Painlevé VI equation  bSpringer3 a<p>I study the solutions of a particular family of Painlevé VI equations with the parameters $\beta=\gamma=0, \delta=1/2$ and $2\alpha=(2\mu-1)^2$, for $2\mu\in\mathbb{Z}$. I show that the case of half-integer $\mu$ is integrable and that the solutions are of two types: the so-called Picard solutions and the so-called Chazy solutions. I give explicit formulae for them and completely determine their asymptotic behaviour near the singular points $0,1,\infty$ and their nonlinear monodromy. I study the structure of analytic continuation of the solutions to the PVI$\mu$ equation for any $\mu$ such that $2\mu\in\mathbb{Z}$. As an application, I classify all the algebraic solutions. For $\mu$ half-integer, I show that they are in one to one correspondence with regular polygons or star-polygons in the plane. For $\mu$ integer, I show that all algebraic solutions belong to a one-parameter family of rational solutions.</p>1 aMazzocco, Marta  uhttp://hdl.handle.net/1963/311801068nas a2200121   4500008004300000245007400043210007000117260001300187520067000200100002000870700002000890856003600910       2000                        en_Ud  00aMonodromy of certain Painlevé-VI transcendents and reflection groups0 aMonodromy of certain PainlevéVI transcendents and reflection gro  bSpringer3 aWe study the global analytic properties of the solutions of a particular family of Painleve\\\' VI equations with the parameters $\\\\beta=\\\\gamma=0$, $\\\\delta={1\\\\over2}$ and $\\\\alpha$ arbitrary. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. This result is used to classify all the algebraic solutions of our Painleve\\\' VI equation.1 aDubrovin, Boris1 aMazzocco, Marta  uhttp://hdl.handle.net/1963/288200388nas a2200109   4500008004100000245007500041210007000116260001000186653002600196100002000222856003600242       1998                        en  d00aAlgebraic Solutions to the Painlevé-VI Equation and Reflection Groups0 aAlgebraic Solutions to the PainlevéVI Equation and Reflection Gr  bSISSA10aPainlevé VI equation1 aMazzocco, Marta  uhttp://hdl.handle.net/1963/557401558nas a2200109   4500008004100000245007100041210006900112260001300181520119800194100002001392856003601412       1997                        en  d00aKam theorem for generic analytic perturbations of the Guler system0 aKam theorem for generic analytic perturbations of the Guler syst  bSpringer3 aWe apply here KAM theory to the fast rotations of a rigid body with a fixed point, subject to a purely positional potential. The problem is equivalent to a small perturbation of the Euler system. The difficulty is that the unperturbed system is properly degenerate, namely the unperturbed Hamiltonian depends only on two actions. Following the scheme used by Arnol\\\'d for the N-body problem, we use part of the perturbation to remove the degeneracy: precisely, we construct Birkhoff normal form up to a suitable finite order, thus eliminating the two fast angles; the resulting system is nearly integrable and (generically) no more degenerate, so KAM theorem applies. The resulting description of the motion is that, if the initial kinetic energy is sufficiently large, then for most initial data the angular momentum has nearly constant module, and moves slowly in the space, practically following the level curves of the initial potential averaged on the two fast angles; on the same time the body precesses around the instantaneous direction of the angular momentum, essentially as in the Euler-Poinsot motion. We also provide two simple physical examples, where the procedure does apply.1 aMazzocco, Marta  uhttp://hdl.handle.net/1963/1038