Dynamics on strata of trigonal Jacobians and some integrable problems of rigid body motion

Abstract

We present an algebraic geometrical and analytical description of the Goryachev case of rigid body motion. It belongs to a family of systems sharing the same properties: although completely integrable, they are not algebraically integrable, their solution is not meromorphic in the complex time and involves dynamics on the strata of the Jacobian varieties of trigonal curves.Although the strata of hyperelliptic Jacobians have already appeared in the literature in the context of some dynamical systems, the Goryachev case is the first example of an integrable system whose solution involves a more general curve. Several new features (and formulae) are encountered in the solution given in terms of sigma-functions of such a curve.1 These can be related to other Jacobians in the case of dimension 2.where J = (J 1 , J 2 , J 3 ) T and γ = (γ 1 , γ 2 , γ 3 ) T are the angular and linear momentum respectively and H(J, γ) is the Hamiltonian, which is also a first integral. In addition to the Hamiltonian the equations always possess the integrals (Casimir functions) C 1 = J, γ , C 2 = γ, γ . Apart from the well known integrable cases of Kirchhoff, Clebsch, Steklov and Lyapunov (and their gyroscopic generalizations) there are some further special cases of integrability cases where an additional integral exists only under the condition C 1 = 0. In the most classical case found by D. Goryachev [Gor912] and S. Chaplygin [Chap904] the extra integral is cubic in J. In this case Chaplygin himself [Chap904] gave a separation of variables and reduced the system to quadratures containing integrals on a hyperelliptic genus 2 curve. A detailed algebro-geometric description of the complex invariant manifolds was made in [BvM987]. Further rather exotic special cases of integrability also exist for which neither separation of variables nor explicit solution were known until recently. Here we concentrate on the Goryachev case [Gor916] which was reduced to quadratures in [VT09] by using a bi-Hamiltonian structure and the corresponding separating Darboux-Nijenhuis variables.2 There are several exceptions in the family: one of them is the classical Goryachev-Chaplygin system [Chap904, Gor916], which is linearized on Jacobians of genus 2 hyperelliptic curve. DYNAMICS ON STRATA OF TRIGONAL JACOBIANS 3 The Goryachev case, the focus of this paper, has Hamiltonian H 1 = H and extra integral H 2 that take the form H 1 = J 2 1 + J 2 2 + 4 3 J 2 3 + aγ 1 + b γ 2/3 3 , a, b being arbitrary constants,

Integrable 2D Time-Irreversible Systems with a Cubic Second Integral

“…This case adds two arbitrary parameters 2 and 3 to the case found by Yehia [21] and has five arbitrary parameters , 0 , 1 , 2 , and 3 , more than the original case found by Goriachev [29] in 1915. Although having no obvious physical meaning, Goriachev's case has received a growing interest in the last years [30,31]. It turns out to be the first example of a mechanical system whose complex invariant varieties are strata of Jacobians of a nonhyperelliptic curve, here a trigonal curve of genus 3 [31].…”

On Addition Formulae for Sigma Functions of Telescopic Curves

“…(ii) For A 3 = (2k, 2k + 2, 2k + 1), k ≥ 2, in (ii) of Example 1, polynomials F i are given by (2) , 2(2k + 1) for (3) .…”

“…It is well known that the solution of Jacobi's inversion problem for a hyperelliptic curve has a simple description by hyperelliptic ℘-functions, the second logarithmic derivatives of the sigma function [2,6]. The inversion of hyperelliptic or more general algebraic integrals of genus g on the Abel-Jacobi image W k of the k-th symmetric products of the curve with k < g is extensively studied in connection with the problem of mathematical physics (see [3,6] and references therein). This problem is intimately related with the problem on the vanishing of the derivatives of the sigma function on W k .…”

“…. , a m ) a sequence of relatively prime positive integers satisfying the telescopic condition (3). Then the telescopic curve associated with (a 1 , .…”

Periods of second kind differentials of $(n,s)$-curves

“…In particular, the approach of Klein (for any Riemann surface of genus 3) and [KSh12] (for an arbitrary Riemann surface of any genus) to the theory of higher genus sigma-functions is based on resolving the generalized Legendre relations in terms of theta-constants. Using (n, s)-curves, it is possible to develop the construction of Abelian functions and the associated integrable PDEs in terms of the σ-function of the trigonal curve, [EEMOP08], [BEGO08], to develop the study of space curves [Mat13], [AN12], to consider τ -functions of integrable hierarchies as σ-functions, [Nak10a], [HE11], to develop the description of classical surfaces like Kummer, Coble surfaces [BEL12], [EGOP13], to describe Jacobi inversion on the strata of non-hyperelliptic Jacobians [MP08], [BEF12], to develop number-theoretical problems [BEH05], [KMP12] and others.…”